The extreme albedo asymmetry of Saturn's moon Iapetus, which is about 10 times as bright on its trailing hemisphere as on its leading hemisphere, has been an enigma for three centuries. Deposition of exogenic dark material on the leading side has been proposed as a cause, but this alone cannot explain the global shape, sharpness, and complexity of the transition between Iapetus' bright and dark terrain. We demonstrate that all these characteristics, and the asymmetry's large amplitude, can be plausibly explained by runaway global thermal migration of water ice, triggered by the deposition of dark material on the leading hemisphere. This mechanism is unique to Iapetus among the saturnian satellites because its slow rotation produces unusually high daytime temperatures and water ice sublimation rates for a given albedo.