SCHEDULE 13G

                       SECURITIES AND EXCHANGE COMMISSION
                              WASHINGTON, DC 20549

                    Under the Securities Exchange Act of 1934
                               (Amendment No. 1)*

                          ALEXION PHARMACEUTICALS, INC.
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                                (Name of Issuer)

                     Common Stock, par value $0.01 per share
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                         (Title of Class of Securities)

                                   015351 10 9
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                                 (CUSIP Number)

                                December 31, 2005
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      (Date of Event which Requires Filing of this Statement) Check the
appropriate box to designate the rule pursuant to which this Schedule is filed:

      |X|   Rule 13d-1(b)

      |_|   Rule 13d-1(c)

      |_|   Rule 13d-1(d)

      *The remainder of this cover page shall be filled out for a reporting
person's initial filing on this form with respect to the subject class of
securities, and for any subsequent amendment containing information which would
alter disclosures provided in a prior cover page.

      The information required on the remainder of this cover page shall not be
deemed to be "filed" for the purpose of Section 18 of the Securities Exchange
Act of 1934 (the "Act") or otherwise subject to the liabilities of that section
of the Act, but shall be subject to all other provisions of the Act (however,
see the Notes).

                                  Page 1 of 7



CUSIP No.  015351 10 9
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  1     NAMES OF REPORTING PERSONS
        I.R.S. IDENTIFICATION NOS. OF ABOVE PERSONS (entities only)

        Felix J. Baker
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  2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (See Instructions)
                                                                         (a) |_|
                                                                         (b) |X|

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  3     SEC USE ONLY
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  4     CITIZENSHIP OR PLACE OF ORGANIZATION

         United States
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       NUMBER OF           5    SOLE VOTING POWER
         SHARES
      BENEFICIALLY                    -0-
        OWNED BY           -----------------------------------------------------
          EACH             6    SHARED VOTING POWER
       REPORTING
         PERSON                        1,360,667
          WITH             -----------------------------------------------------
                           7    SOLE DISPOSITIVE POWER

                                          -0-
                           -----------------------------------------------------
                           8    SHARED DISPOSITIVE POWER

                                        1,360,667
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  9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

        1,360,667
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  10    CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES
        (See Instructions)
                                                                             |_|
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  11    PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

        4.2%
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  12    TYPE OF REPORTING PERSON (See Instructions)

        IN
--------------------------------------------------------------------------------

                                  Page 2 of 7



CUSIP No.  015351 10 9
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  1     NAMES OF REPORTING PERSONS
        I.R.S. IDENTIFICATION NOS. OF ABOVE PERSONS (entities only)

        Julian C. Baker
--------------------------------------------------------------------------------

  2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (See Instructions)

                                                                         (a) |_|
                                                                         (b) |X|
--------------------------------------------------------------------------------

  3     SEC USE ONLY
--------------------------------------------------------------------------------

  4     CITIZENSHIP OR PLACE OF ORGANIZATION

        United States
--------------------------------------------------------------------------------
     NUMBER OF        5    SOLE VOTING POWER
      SHARES
   BENEFICIALLY            -0-
     OWNED BY         ----------------------------------------------------------
       EACH
     REPORTING        6    SHARED VOTING POWER
      PERSON
       WITH                1,360,667
                      ----------------------------------------------------------

                      7    SOLE DISPOSITIVE POWER

                           -0-
                      ----------------------------------------------------------

                      8    SHARED DISPOSITIVE POWER

                           1,360,667
--------------------------------------------------------------------------------

  9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

        1,360,667
--------------------------------------------------------------------------------

  10    CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES
        (See Instructions)
                                                                             |_|
--------------------------------------------------------------------------------

  11    PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

        4.2%
--------------------------------------------------------------------------------

  12    TYPE OF REPORTING PERSON (See Instructions)

        IN

--------------------------------------------------------------------------------

                                  Page 3 of 7



Item 1(a)         Name of Issuer:

                  Alexion Pharmaceuticals, Inc.

Item 1(b)         Address of Issuer's Principal Executive Offices:

                  352 Knotter Drive
                  Cheshire, Connecticut  06410

Item 2(a)         Name of Person Filing:

                  This Schedule 13G is being filed jointly by Felix J. Baker and
                  Julian C. Baker (the "Reporting Persons").

Item 2(b)         Address of Principal Business Office or, if None, Residence:

                  Name                  Business Address
                  Felix J. Baker        667 Madison Avenue
                                        New York, NY 10021
                  Julian C. Baker       667 Madison Avenue
                                        New York, NY 10021
Item 2(c)         Citizenship:

                  Each of the Reporting Persons is a United States citizen.

Item 2(d)         Title of Class of Securities:

                  Common Stock, par value $0.01 per share

Item 2(e)         CUSIP Number:

                  015351 10 9

Item 3.           If this statement is filed pursuant to ss.ss.240.13d-1(b) or
                  (c), check whether the person filing is a:   N/A

      (a)   [ ] Broker or dealer registered under Section 15 of the Exchange
            Act;

      (b)   [ ] Bank as defined in section 3(a)(6) of the Exchange Act;

      (c)   [ ] Insurance company as defined in section 3(a)(19) of the Exchange
            Act;

      (d)   [ ] Investment company registered under section 8 of the Investment
            Company Act of 1940;

      (e)   [ ] An investment adviser in accordance with Rule
            13d-1(b)(1)(ii)(E);

                                  Page 4 of 7



      (f)   [ ] An employee benefit plan or endowment fund in accordance with
            Rule 13d-1(b)(1)(ii)(F);

      (g)   [ ] A parent holding company or control person in accordance with
            Rule 13d-1(b)(1)(ii)(G).

      (h)   [ ] A savings association as defined in Section 3(b) of the Federal
            Deposit Insurance Act.

      (i)   [ ] A church plan that is excluded from the definition of an
            investment company under section 3(c)(14) of the Investment Company
            Act of 1940.

      (j)   [ ] Group, in accordance with Rule 13d-1(b)(1)(ii)(J).

Item 4. Ownership.

      Set forth below is the aggregate number of shares of Common Stock,
including shares that may be acquired upon conversion of 1.375% Convertible
Notes due February 2012 at the presently applicable conversion price of
$31.7914, held as of the December 31, 2005 by each of the following, together
with the percentage of the outstanding shares of Common Stock that such number
represents based upon 30,901,000, as reported on the Company's Form 10-Q for the
quarterly period ended October 31, 2005.

                                                             Percent of Class
Name                                      Number of Shares      Outstanding
----                                      ----------------      -----------
Baker Bros. Investments, L.P.                     41,455               0.1%
Baker Bros. Investments II, L.P.                  37,323               0.1%
Baker Biotech Fund I, L.P.                       410,871               1.3%
Baker Biotech Fund II, L.P.                      374,757               1.2%
Baker Biotech Fund II (Z), L.P.                   49,213               0.2%
Baker Biotech Fund III, L.P.                     316,928               1.0%
Baker Biotech Fund III (Z), L.P.                  67,461               0.2%
14159, L.P.                                       27,054               0.1%
Baker/Tisch Investments, L.P.                     35,605               0.1%
                                               ---------               ---
Total                                          1,360,667               4.2%
                                               =========               ===

      By virtue of their ownership of entities that have the power to control
the investment decisions of the limited partnerships listed in the table above,
Felix J. Baker and Julian C. Baker may each be deemed to be beneficial owners of
securities owned by such entities and may be deemed to have shared power to vote
or direct the vote of and shared power to dispose or direct the disposition of
such securities.

Item 5. Ownership of Five Percent or Less of a Class.

      If this statement is being filed to report the fact that as of the date
hereof the reporting person has ceased to be the beneficial owner of more than
five percent of the class of securities, check the following [ X ].

Item 6. Ownership of More than Five Percent on Behalf of Another Person.

      The entities listed in Item 4 above are investment funds the investors in
which have the right to receive dividends, interest and the proceeds of sale of
securities owned by such funds.

Item 7. Identification and Classification of the Subsidiary Which Acquired the
Security Being Reported on by the Parent Holding Company or Control Person.

      N/A

Item 8. Identification and Classification of Members of the Group.

      N/A

Item 9. Notice of Dissolution of Group.

      N/A

Item 10. Certification.

      By signing below I certify that, to the best of my knowledge and belief,
the securities referred to above were not acquired and are not held for the
purpose of or with the effect of changing or influencing the control of the
issuer of the securities and were not acquired and are not held in connection
with or as a participant in any transaction having that purpose or effect.

                                  Page 5 of 7



                                    SIGNATURE

      After reasonable inquiry and to the best of my knowledge and belief, I
hereby certify that the information set forth in this statement is true,
complete and correct.

February 14, 2006

                                        /s/ Felix J. Baker
                                        ----------------------------------
                                        Felix J. Baker


                                        /s/ Julian C. Baker
                                        ----------------------------------
                                        Julian C. Baker

                                  Page 6 of 7



                                    AGREEMENT

      In accordance with Rule 13d-1(k)(1) under the Securities Exchange Act of
1934, as amended, the undersigned hereby agree that this Statement on Schedule
13G relating to the Common Stock of Alexion Pharmaceuticals, Inc. is being filed
with the Securities and Exchange Commission on behalf of each of them.

February 14, 2006

                                        /s/ Felix J. Baker
                                        ----------------------------------
                                        Felix J. Baker


                                        /s/ Julian C. Baker
                                        ----------------------------------
                                        Julian C. Baker

                                  Page 7 of 7