**Exhibit 31.1****Section 302 Certification**I, David W. Sharp, certify that:1. I have reviewed this Quarterly Report on Form 10-Qfor the quarter ended March 31, 2006 ofHorizon Offshore, Inc.;2. Based on my knowledge, this report does not containany untrue statement of a material fact oromit to state a material fact necessary to make the statements made, in light of the circumstancesunder which such statements were made, not misleading with respect to the period covered by thisreport;3. Based on my knowledge, the financial statements, andother financial information included inthis report, fairly present in all material respects the financial condition, results of operationsand cash flows of the registrant as of, and for, the periods presented in this report;4. The registrant’s other certifying officer(s) and Iare responsible for establishing andmaintaining disclosure controls and procedures (as defined in Exchange Act Rules 13a-15(e) and15d-15(e)) and internal control over financial reporting (as defined in Exchange Act Rules13a-15(f) and 15d-15(f)) for the registrant and have:     (a) Designedsuch disclosure controls and procedures, or caused suchdisclosure controls and procedures to be designed under oursupervision, to ensure that material information relating to theregistrant, including its consolidated subsidiaries, is made knownto us by others within those entities, particularly during theperiod in which this report is being prepared;     (b) Designed suchinternal control over financial reporting, orcaused such internal control over financial reporting to be designedunder our supervision, to provide reasonable assurance regarding thereliability of financial reporting and the preparation of financialstatements for external purposes in accordance with generallyaccepted accounting principles;     (c) Evaluated theeffectiveness of the registrant’s disclosurecontrols and procedures and presented in this report our conclusionsabout the effectiveness of the disclosure controls and procedures,as of the end of the period covered by this report based on suchevaluation; and     (d) Disclosedin this report any change in the registrant’s internalcontrol over financial reporting that occurred during theregistrant’s most recent fiscal quarter (the registrant’s fourthfiscal quarter in the case of an annual report) that has materiallyaffected, or is reasonably likely to materially affect, theregistrant’s internal control over financial reporting; and5. The registrant’s other certifying officer(s) and I havedisclosed, based on our most recentevaluation of internal control over financial reporting, to the registrant’s auditors and the auditcommittee of the registrant’s board of directors (or persons performing the equivalent functions):     (a) All significantdeficiencies and materialweaknesses in the design or operation of internalcontrol over financial reporting which arereasonably likely to adversely affect theregistrant’s ability to record, process, summarizeand report financial information; and     (b) Any fraud,whether or not material, thatinvolves management or other employees who have asignificant role in the registrant’s internalcontrol over financial reporting.             /s/ David W. Sharp    David W. Sharp   *President and Chief Executive Officer*