----

**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; |
|----|-----------------------------------------------------------------------------------------------------------------------|

| 2. | Based                on my knowledge, this 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                report; |
|----|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

| 3. | Based                on my knowledge, the financial statements, and other financial information                included in this 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                report; |
|----|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

| 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: |
|----|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | 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; |
|--------|----|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | 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; |
|--------|----|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | 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 |
|--------|----|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | 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 |
|--------|----|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

| 5. | The                registrant's other certifying officer(s) 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 the registrant's                board of directors (or persons performing the equivalent                functions): |
|----|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | 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 |
|--------|----|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | 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. |
|--------|----|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|

|   | /s/                BEN M. BRIGHAM                                           |   |
|--------|-----------------------------------------------------------------------------|--------|
|   | Ben                M. Brigham                                               |   |
|   | Chief                Executive Officer, President and Chairman of the Board |   |

----