The student pilot planned to conduct a solo cross-country flight to another towered airport. His preflight inspection, engine start and runup, takeoff, and flight of the special light-sport airplane were all uneventful. When he neared the destination airport, he was instructed to enter the traffic pattern on the downwind leg for runway 16. He was subsequently cleared for "closed traffic" pattern work, and instructed to follow another airplane. The pilot informed the air traffic controller that he intended to do a full stop landing, and shortly thereafter, was instructed to turn base leg. He selected full flaps (40 degrees) for landing after turning final. According to the pilot, the wind was from 180 degrees at 7 knots, and the airplane was "just a little" to the right of the centerline for part of the last segment of the final approach. The pilot flared for landing, and the airplane touched down on the main gear, but then "ballooned" back airborne. The airplane bounced again, and then on the third bounce, the nose landing gear failed. The airplane slid to a stop on the runway. The airplane sustained substantial damage to the fuselage structure. During the bounce sequence, the pilot only used the control wheel to attempt to control and stop the bounces; he did not vary the throttle setting. The pilot reported no preimpact mechanical malfunctions or failures with the airplane that would have precluded normal operation.