EXHIBIT 32.1

Certification of CEO Pursuant to

18 U.S.C. Section 1350,

as Adopted Pursuant to

Section 906 of the Sarbanes-Oxley Act of 2002

In connection with the Annual Report on Form 10-K of Ambac Financial Group, Inc. (the “Company”) for the period ended December 31, 2003, as filed with the Securities and Exchange Commission on the date hereof (the “Report”), Robert J. Genader, as Chief Executive Officer of the Company, hereby certifies, pursuant to 18 U.S.C. § 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that, to the best of his knowledge:

(1) The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

(2) The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

/s/    ROBERT J. GENADER        

Name:  Robert J. Genader

Title:    Chief Executive Officer and President

Date:    March 15, 2004