After descending to 400 feet above mean sea level along a beach on the return leg of a cross country flight, the pilot attempted to increase power to level off but, the engine would not respond. The pilot then checked to make sure the fuel valve was on "BOTH," and the primer was in and locked. He then attempted to restart the engine. The engine however, would not restart. The pilot determined that the beach was the best and safest place to land, so he extended the wing flaps to the full down position, and landed on the beach. During the landing, the nose landing gear dug into the sand, and collapsed. Examination of the airplane revealed that it was substantially damaged and approximately 12 gallons of fuel was still on board. Further examination revealed no evidence of any preimpact mechanical failure or malfunctions of the engine that would have precluded normal operation. After the wreckage was recovered the engine was started and run at full power. Review of a carburetor icing chart revealed that atmospheric conditions around the time of the accident were conducive to icing at glide and cruise power. When asked if he had used carburetor heat during his descent prior to the loss of engine power, the pilot advised that he had not used it.