The pilot reported that, on a previous flight, the airplane had been filled with fuel and flown for about 3.8 hours. He added that he attempted to refuel the airplane for the accident flight but that the self-service fuel station at the departure airport "failed to dispense fuel." He stated that he visually checked the tanks and calculated that there would be enough fuel to make the flight. He added that, after departing, during cruise flight about 1,800 ft above ground level, the "engine stopped" and that he then set the "best glide speed" toward a nearby airport. After realizing the airplane would not reach the airport, he executed a 180º turn and landed the airplane on a snow-covered, frozen lake with the landing gear retracted. The airplane sustained substantial damage to the left wing and fuselage. The pilot reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation.