During flight, the instructor asked the student to perform a power-off approach-to-landing stall. The student began the maneuver at about 4600 ft with the carburetor heat on. During recovery, he abruptly advanced the throttle and the engine began to lose power. The instructor took control of the aircraft, and shortly thereafter, the engine ceased operating. The instructor attempted to restart the engine, but said that he was unable, due to lack of battery power. During a subsequent forced landing, the aircraft touched down on terrain covered with deep snow and nosed over. During an investigation, the carburetor and associated components were inspected for ice or other foreign material; however, none was found. The engine was removed from the aircraft and placed on a test stand. It operated normally during a ground run. Previously, one cell of the battery would not hold a charge. This required "hand propping" of the engine on the first flight of the day. Also, the battery was slow to recharge.