OBO, Central African Republic—Somewhere in dense savannas and jungles spanning four African nations is Joseph Kony, a self-proclaimed prophet and commander of the Lord's Resistance Army. On his trail are thousands of Ugandan troops, smaller forces from nearby countries and the U.S. Green Berets.

The Americans joined the hunt about six months ago to track a man whose followers have cut a swath of violence through the heart of Africa—leaving rape victims, child soldiers, mutilated bodies and destabilized governments in their...