On February 3, 1998, at 0900 central standard time, a Piper PA-22-108 airplane, N4654Z, was substantially damaged during a runway overrun while landing near Frederick, Oklahoma. The student pilot, sole occupant of the airplane, was not injured. The airplane was owned and operated by the pilot under Title 14 CFR Part 91. Visual meteorological conditions prevailed for the instructional flight for which a flight plan was not filed. The local flight originated from the Frederick Municipal Airport at approximately 0800. The student pilot reported that the flight departed from Runway 35 at the Frederick Municipal Airport "for some upper air work in the local practice area." The pilot further stated that fog started to form in the vicinity of the airport, so he elected to proceed direct to the airport to avoid the inclement weather. The pilot stated that he was aware that he was landing downwind on runway 21, but feared that he might have lost visual contact with the airport if he had attempted to circle the airport to land on runway 03. The pilot reported the winds from 030 degrees at 15 knots. The pilot stated that he made a normal touchdown midfield on the 4,800 foot concrete runway. During the landing roll, he started to apply the wheel brakes as he approached the departure end of the runway. While applying brake pressure with the brake lever, the cable to the brakes snapped, resulting in the total loss of braking action. The airplane overran the departure end of the runway, and the nose wheel impacted a ridge/terrace in the wheat field adjacent to the airport. The student pilot had accumulated a total of 35 hours, of which 28 were in the accident airplane. He reported that he purchased the airplane in September from its second owner. The airplane had accumulated 12 hours since it's last annual inspection, which was completed in November 1997. The pilot suspected that the cable that failed was the one that came with the airplane when it was manufactured. The nose gear dug into the soft ridge, and the airplane nosed over coming to rest in the inverted position. The vertical fin, rudder, the struts for the right wing, and the main spar for the left wing sustained structural damage. The leading edge of the right wing was also damaged. The 1961 model "Colt" had accumulated a total of 1,715 airframe hours. According to the manufacturer's illustrated parts catalog, the failed brake cable assembly, part number 40123-91, is a condition item without a specific replacement or retirement time. The cable separated at the point where it makes a sharp turn over a pulley (part number 11200-00).