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**Exhibit      31.1**

**CERTIFICATION      OF CHIEF EXECUTIVE OFFICER**

**PURSUANT      TO RULE 13a-14(a) OF THE **

**SECURITIES      EXCHANGE ACT OF 1934**

| 1. | I                  have reviewed this quarterly report on Form 10-Q of Brigham                  Exploration Company; |
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| 2. | Based                  on my knowledge, this annual report does not contain any untrue                  statement                  of a material fact or omit to state a material fact necessary to                  make the                  statements made, in light of the circumstances under which such                  statements                  were made, not misleading with respect to the period covered by                  this                  annual report; |
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| 3. | Based                  on my knowledge, the financial statements, and other financial                  information                  included in this annual report, fairly present in all material                  respects                  the financial condition, results of operations and cash flows of                  the                  registrant as of, and for, the periods presented in this annual                  report; |
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| 4. | The                  registrant's other certifying officer and I are responsible for                  establishing and maintaining disclosure controls and procedures                  (as                  defined in Exchange Act Rules 13a-15(e) and 15d-15(e)) and                  internal                  control over financial reporting (as defined in Exchange Act Rules                  13a-15(f) and 15d-15(f)) for the registrant and                  have: |
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|   | a) | Designed                  such disclosure controls and procedures or caused such disclosure                  controls                  and procedures to be designed under our supervision, to ensure                  that                  material information relating to the registrant, including its                  consolidated subsidiaries, is made known to us by others within                  those                  entities, particularly during the period in which this report is                  being                  prepared; |
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|   | b) | Designed                  such internal control over financial reporting, or caused such                  internal                  control over financial reporting to be designed under our supervision,                  to                  provide reasonable assurance regarding the reliability of financial                  reporting and the preparation of financial statements for external                  purposes in accordance with generally accepted accounting                  principles; |
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|   | c) | Evaluated                  the effectiveness of the registrant's disclosure controls and procedures                  and presented in this report our conclusions and about the effectiveness                  of the disclosure controls and procedures, as of the end of the                  period                  covered by this report based on such evaluation;                  and |
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|   | d) | Disclosed                  in this report any change in the registrant’s internal control over                  financial reporting that occurred during the registrant’s most recent                  fiscal quarter (the registrant’s fourth fiscal quarter in the case of an                  annual report) that has materially affected, or is reasonably likely                  to                  materially affect, the registrant’s internal control over financial                  reporting; and |
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| 5. | The                  registrant's other certifying officer and I have disclosed, based                  on our                  most recent evaluation of internal control over financial reporting,                  to                  the registrant's auditors and the audit committee of registrant's                  board of                  directors (or persons performing the equivalent                  function): |
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|   | a) | All                  significant deficiencies and material weaknesses in the design                  or                  operation of internal control over financial reporting, which are                  reasonably likely to adversely affect the registrant's ability                  to record,                  process, summarize and report financial information;                  and |
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|   | b) | Any                  fraud, whether or not material, that involves management or other                  employees who have a significant role in the registrant's internal                  control                  over financial reporting. |
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|   | /s/                BUD                M.                BRIGHAM             |   |
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|   | Bud                M. Brigham                                               |   |
|   | Chief                Executive Officer, President and Chairman of the Board |   |

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