On June 3, 2013, about 1520 eastern daylight time, a Piper PA-34-200T, N633TC, experienced a trim cable failure during climbout from Tallahassee Regional Airport (TLH), Tallahassee, Florida. The flight instructor (CFI) and student pilot were not injured, and the airplane was not damaged. Visual meteorological conditions prevailed and no flight plan was filed. The airplane was registered to and operated by Florida Flight Training Center under the provisions of Title 14 Code of Federal Regulations Part 91 as an instructional flight. The flight was originating at the time of the incident. The CFI stated they departed from TLH uneventfully and were climbing through 300 feet when the student pilot noticed a problem with the elevator trim wheel. The CFI moved the trim wheel and noted no resistance when the wheel was turned. Upon looking down at the wheel, he observed the trim cable protruding out from the side of the wheel. The student pilot advised that he was having difficulty controlling the airplane and the CFI subsequently assumed control of the airplane. The CFI advised the tower controller of the emergency and landed the airplane without incident. Examination of the airplane revealed that the elevator trim cable had broken and was protruding from the trim wheel housing. The elevator trim cable assembly was sent to the NTSB material laboratory for examination. An examination of the trim cable revealed that it was composed of a 1/16 inch diameter wire rope with a swaged-on ball at one end and a swaged-on turnbuckle fitting at the other end. Visual inspection revealed trim cable lubrication along the entire length with no indication of external corrosion. The wire rope section had separated about 5 feet 6 inches from the turnbuckle end and 14 feet 6 inches from the ball end of the cable. The separation was made up of individual wire breaks spread along an approximate 2-inch length of the wire rope. A magnified visual inspection of the cleaned cable established that all but a few wires were fractured directly across the wires with no apparent yielding or deformation. Further, only a few wires (broken or intact) showed either internal or external wear. A scanning electron microscope examination of several of the broken wires revealed features consistent with fatigue progression through the individual wires. A review of the aircraft logbooks revealed that the failure of the trim cable was not detected on April 25, 2013, -during the 100 hour and annual inspections. Nor was it noted on April 13, 2013, during the compliance of AD 2013-02-13; which states in section 7-149 a) "At each annual or 100 hour inspection, all control cables must be inspected for broken wires strands. Any cable assembly that has one broken wire strand located in a critical fatigue area must be replaced." The inspection was completed 3 months prior to this event.