At about 1236 Eastern time, the airplanes nose landing gear collapsed while landing. Visual conditions prevailed; the Commercial pilot was not injured. While on final approach, the pilot confirmed the landing gear was down and locked. When the flight was approximately 10 to 15 feet agl, he reduced power and applied nose-up trim using the electric pitch trim system. The pitch trim kept running in the nose-up direction, which he corrected by applying power and forward pressure on the control yoke. The airplane touched down on the runway on all three landing gear, bounced and then landed on the nose landing gear, which collapsed approximately one second later. Postaccident examination revealed the stabilator trim would continue to operate in the nose-down direction, even though the electric pitch trim switch was released.