Exhibit 32.1

CERTIFICATIONS PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

The undersigned officer of ACE Limited (the “Corporation”) hereby certifies that the Corporation’s Quarterly Report on Form 10-Q for the quarter ended June 30, 2004, fully complies with the applicable reporting requirements of Section 13(a) of the Securities Exchange Act of 1934 (15 U.S.C. 78m(a)) and that the information contained in the Form 10-Q fairly presents, in all material respects, the financial condition and results of operations of ACE Limited.

Dated: August 6, 2004	/S/    EVAN GREENBERG
	Evan Greenberg President and Chief Executive Officer