EXHIBIT 32.1

CERTIFICATION PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

I, I. Jon Brumley, Chief Executive Officer of Encore Acquisition Company (the “Company”), in connection with the filing of the Company’s Annual Report of Company on Form 10-K with the Securities and Exchange Commission on the date hereof (the “Report”), for the year ended December 31, 2004, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, certify that:

      (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 result of operations of the Company.

/s/ I. Jon Brumley

I. Jon Brumley

Chief Executive Officer

March 9, 2005

      A signed original of this written statement required by Section 906 has been provided to the Company and will be retained by the Company and furnished to the Securities and Exchange Commission or its staff upon request.

      This certification accompanies the report pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 and shall not be deemed filed by the Company for purposes of Section 18 of the Exchange Act of 1934, as amended.