According to the student pilot, she had just landed after a local instructional flight. Her instructor completed the solo endorsement in her logbook as they taxied back to the arrival end of the runway 28. The student pilot then taxied on to runway 28 for departure. She applied full throttle and the airplane lifted off the runway normally. At approximately 500 feet above ground level the engine began to run rough and dropped to 1900 RPMs. The student pilot immediately lowered the nose of the airplane to attain best glide speed and started a left hand turn to return to the airport. She attempted to land on runway 10. Approximately 20 seconds after the engine began running rough the pilot applied the carburetor heat and then reduced the throttle to idle. The student pilot did not attempt however to increase power after the application of carburetor heat and the throttle remained in the idle position for the remainder of the approach. The pilot then applied full flaps as the airplane was descending over midfield. The airplane overflew the runway and touched down in the grass immediately off the departure end of runway 10. The airplane subsequently struck a fence and nosed over. Post accident examination of the airplane and engine showed no evidence of mechanical failure or malfunction. The engine was started and operated with no signs of failure. A review of the carburetor icing probability chart shows the aircraft was operating in conditions that were not conductive to carburetor icing at the time of the accident. The pilot had 43 total flight hours.