On July 12, 1999, about 1225 eastern daylight time, a Cessna 172H, N3213L, was substantially damaged during a forced landing at the Gene Snyder Airport, Falmouth, Kentucky. The certificated commercial pilot and the passenger received minor injuries. Visual meteorological conditions prevailed for the personal flight that originated from Huntington, West Virginia, destined for Mount Vernon, Illinois. A visual flight rules flight plan was filed, and the flight was conducted under 14 CFR Part 91. According to the pilot, the day before the accident, he went to the airport about 0800, preflighted the airplane, then stared the airplane's engine. The engine ran "rough", so he aborted the flight. That same day, the pilot had a certificated mechanic inspect the engine. The mechanic thought the problem might be a stuck valve, so the next day, the pilot had a maintenance shop at the airport examine the engine. The shop identified and repaired a stuck valve on the number one cylinder, and then ran the engine. The engine ran "smoothly," and the pilot accepted the airplane. After paying the maintenance facility, the pilot checked the weather, and filed a flight plan. He conducted a preflight, and noticed no anomalies. He started the airplane's engine, and taxied for fuel. After filling both tanks, the pilot boarded his passenger, and taxied to runway 12 for an intersection departure at taxiway "A." Once on the runway, the pilot advanced the throttle, and the engine responded "smoothly." He executed the takeoff, turned to a heading of about 315 degrees magnetic, and climbed to 3,500 feet msl. While en route, the pilot activated his flight plan, and requested flight following from Indianapolis Center. Approximately 45 minutes after departure, the engine started running "rough." The pilot repositioned the fuel selector, checked both magnetos, and selected carburetor heat-on. Nothing changed. With the engine continuing to run "rough", the pilot squawked 7700 on the transponder, and turned towards Gene Snyder Airport. He attempted to contact Indianapolis Center for assistance, but was unable. He then contacted Louisville Radio, which relayed radar vectors from center. The pilot established the airplane on extended final for runway 21, but aborted the approach approximately 1/2 mile from the airport because of a suspected tailwind condition. Initially, the airplane climbed after aborting the approach, but then started to descend again. The pilot continued on runway heading in an attempt to identify the winds. Approximately half way down the 4,000 foot runway and 500 feet agl, the pilot experienced a further loss in engine power. The pilot selected carburetor heat-off, but power required still exceeded power available for level flight. With insufficient runway remaining to land, and insufficient power to maneuver for runway 21, the pilot turned left, hoping to identify an off-airport landing site. He found nothing, so he continued the left turn. While clearing trees by approximately 25 feet, the pilot identified an area of small trees on the far side of the runway. He maneuvered the airplane towards the trees. Before crossing perpendicular to the runway at 50 feet agl, the pilot experienced a total loss of engine power. He secured the magnetos, and attempted to set the fuel selector to off prior to impact. The airplane impacted the trees flaps up, and indicating approximately 70 mph. After the airplane came to rest, the pilot saw fuel leaking from the airplane. Fearing a post crash fire, he unsuccessfully made a second attempt to secure the fuel selector, then egressed. In addition, the pilot stated that prior to the engine running "rough," the exhaust gas temperature (EGT) gauge was indicating "normal." Afterwards, EGT dropped to below "normal." The EGT sensor was located on the number one cylinder. According to a Federal Aviation Administration (FAA) Inspector, he found no anomalies with the engine during his examination, except for a broken intake valve rocker arm on the number one cylinder. According to another FAA Inspector, the maintenance facility that repaired the stuck valve prior to the accident flight placed a rope in the cylinder via the sparkplug hole, and then brought the piston up against the rope, dislodging the stuck valve. The intake valve rocker-arm was sent to the Safety Board's Materials Laboratory in Washington D.C. for examination. During the examination, no evidence of fatigue or any other form of progressive crack growth was observed on either half of the fractured rocker arm.