EXHIBIT 32.1

CERTIFICATION PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

(18 U.S.C. SECTION 1350)

     In connection with the accompanying Annual Report of Brigham Exploration Company (the “Company”) on Form 10-K, for the period ended December 31, 2003 (the “Report”), I, Ben M. Brigham, Chief Executive Officer, President and Chairman of the Board of the Company, hereby certify that:

(1)     The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. 78m or 78o(d)); and

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

Dated: March 26, 2004

/s/ Ben M. Brigham

Ben M. Brigham

Chief Executive Officer