At the end of a 2-hour flight the airplane was 3.8 miles from the Cheyenne Airport when the engine began to surge. The pilot overflew the airport in order to troubleshoot the problem. The pilot reported that he was at 135 knots, and the fuel selector was set to the right fuel tank, which had approximately 10 gallons of fuel in it. The surging stopped, and the pilot requested a left downwind for runway 27. Within 1 minute the surging returned, he turned toward the airfield, and the engine lost all power. The pilot maintained 125 knots and performed a dead stick landing on runway 31. The airplane veered off the side of the runway and into flat unprepared terrain and the gear collapsed. The builder of the airplane related to the pilot that the SX300 has very limited wing dihedral, and the ball needs to be kept in the center, or else the fuel pickup would unport. The pilot did not report finding any other anomalies with the airframe or engine.