Exhibit 99.1

NOT FOR IMMEDIATE RELEASE – Draft

Contacts:

Alexion Pharmaceuticals, Inc. Leonard Bell, M.D. Chief Executive Officer (203) 272-2596 Euro RSCG Life NRP Robert Stanislaro (Media) (212) 845-4268 Rx Communications Rhonda Chiger (Investors) (917) 322-2569

ALEXION PHARMACEUTICALS COMPLETES OFFERING OF CONVERTIBLE

SENIOR NOTES RAISING $150 MILLION

CHESHIRE, Conn., January 25, 2005 — Alexion Pharmaceuticals, Inc. (Nasdaq: ALXN) today announced it has completed the previously announced sale of $150 million principal amount of its 1.375% Convertible Senior Notes due 2012 to qualified institutional buyers pursuant to Rule 144A under the Securities Act of 1933, as amended. The closing included the exercise in full by the initial purchasers of their option to purchase an additional $25 million principal amount of notes. The Notes are convertible into Alexion’s common stock at an initial conversion rate of 31.7914 shares of common stock per $1,000 principal amount of notes, subject to adjustment (equivalent to a conversion price of approximately $31.46 per share and a conversion premium of approximately 35% to the last reported sale price of $23.30 per share on January 19, 2005).

The net proceeds from this offering will be applied to redeem Alexion’s outstanding $120 million principal amount of 5¾% Convertible Subordinated Notes due 2007 and for general corporate purposes.

The notes to be offered and the common stock issuable upon conversion of the notes have not been registered under the Securities Act of 1933, as amended, or any state securities laws, and may not be offered or sold in the United States absent registration or an applicable exemption from registration requirements.

This press release does not constitute an offer to sell, or the solicitation of an offer to buy, any security and shall not constitute an offer, solicitation or sale in any jurisdiction in which such offering would be unlawful.

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