On an ILS approach to runway 36, the airplane disappeared off radar at 2,500 ft MSL about 12 nautical miles south of the airport. During the approach, the airplane was west of the localizer and was being vectored to intercept the localizer. A complete left 360 degree turn was made at the request of approach control (ARR). The pilot was then cleared to the ILS outer marker. Upon arrival to the outer marker the pilot was told to contact the local controller (LC). Before intercepting the localizer, the airplane started an approximate 9.5 degree left turn away from the localizer. During this turn, ARR queried the pilot about the airplane's heading. There was no response. The LC broadcasted to the airplane and requested the pilot to switch back to ARR. The pax acknowledged the request. There were no further transmissions by the pilot or pax. The investigation did not disclose any evidence of structural, instrument, or engine failure.