PROSPECTUS SUPPLEMENT
(To prospectus dated July 8, 1997)

                                 [ALEXION LOGO]

                          Alexion Pharmaceuticals, Inc.

                        1,974,744 Shares of Common Stock

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

      This document supplements the prospectus dated July 8, 1997 relating to
the sale of 1,974,744 shares of our common stock. We are not selling any shares;
the stockholders named in the prospectus are offering their shares. This
prospectus supplement is incorporated by reference into the prospectus.

                Our Nasdaq National Market Trading Symbol - ALXN

                        Closing Price (June 14, 1999): 10

      Caduceus Capital, Ltd. and Caduceus Capital, L.P., two of the selling
stockholders named in the prospectus, have transferred their securities
portfolios to be managed and traded by a new trust, Caduceus Capital Trust. This
supplement amends the table on page 11 of the prospectus to remove Caduceus
Capital Ltd. and Caduceus Capital, L.P. as selling stockholders from the table
and to add Winchester Global Trust, as Trustee for Caduceus Capital Trust, as a
selling stockholder owning, and offering hereby, a total of 40,000 shares of our
common stock, as described on the following page.

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

      See "Risk Factors," which begins of page 4 of the accompanying prospectus,
for certain information that should be considered by prospective investors.

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

      Neither the Securities and Exchange Commission nor any state securities
commission has approved or disapproved these securities, or determined if this
prospectus is truthful or complete. Any representation to the contrary is a
criminal offense.

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

            The date of this prospectus supplement is June 15, 1999.

- ----------------------------------------------------------------------------------------------------- Amount of Amount of Beneficial Beneficial Ownership Ownership Prior to Offering After Offering - ----------------------------------------------------------------------------------------------------- Number of Name of Number Percent Shares Being Number Percent Selling Stockholder(1) of Shares(2) of Class Offered of Shares of Shares - ----------------------------------------------------------------------------------------------------- Winchester Global Trust, as Trustee 40,000 * 40,000 0 0 for Caduceus Capital Trust (12) - ----------------------------------------------------------------------------------------------------- - ---------- (12) Winchester Global Trust is the sole trustee of Caduceus Capital Trust. Caduceus Capital Trust manages and trades the securities portfolios of Caduceus Capital, L.P. and Caduceus Capital, Ltd. Caduceus Capital Trust has sole voting and dispositive power over the shares offered hereby. Orbimed Advisors LLC is the general partner of and financial adviser to Caduceus Capital, L.P. and an investor in Caduceus Capital, Ltd. Samuel D. Isaly is the managing member of Orbimed Advisors LLC. Sven Borho, Arvind Desai, Carl Gordon and Michael Sheffery are members of Orbimed Advisors LLC. As such, Winchester Global Trust, Caduceus Capital, L.P., Caduceus Capital, Ltd., Orbimed Advisors LLC and Messrs. Isaly, Borho, Desai, Gordon and Sheffery may be deemed beneficial owners of these shares.