On August 19, 1993, about 0630 hours mountain standard time, a Piper PA-28-180, N9533J, crashed during a forced landing about 24 miles northeast of Showlow, Arizona. The airplane was being operated as a visual flight rules (VFR) cross country instructional flight to Tucumcari, New Mexico, when the accident occurred. The airplane, co-owned and operated by the pilot, received substantial damage. The certificated commercial pilot/certified flight instructor (first pilot) and a non-certified student pilot (second pilot) were not injured. Visual meteorological conditions prevailed. A VFR flight plan was filed. The flight originated at Payson, Arizona, at 0510 hours. The first pilot reported that while in cruise flight at 11,500 feet mean sea level the engine suddenly began to run rough and then quit. The pilot adjusted the mixture control in an attempt to restart the engine; however, movement of the control did not produce any results. The second pilot, who was initially flying the airplane, selected a clear area of high desert terrain for a forced landing. Just before touchdown, the first pilot took the controls and completed the landing. During the landing roll, the airplane collided with large bushes and high vegetation and received damage to the nose, landing gear, and wings. Mr. John Eller, Airworthiness Inspector, Federal Aviation Administration, Scottsdale Flight Standards District Office, examined the mixture control cable. He reported that the balljoint assembly that attaches the control cable to the mixture control arm on the carburetor had separated. The assembly, part number 31747.00, was not recovered. The airplane and engine had accumulated a total time in service of 2,545.60 flight hours. Examination of the maintenance records revealed that the most recent annual inspection was accomplished on August 6, 1993, 10 flight hours before the accident. The engine had accrued 420.5 hours of operation since being overhauled.