When a drop of a volatile liquid is deposited on a uniformly heated wettable, thermally conducting substrate, one expects to see it spread into a thin film and evaporate. Contrary to this intuition, due to thermal Marangoni contraction, the deposited drop contracts into a spherical-cap-shaped puddle, with a finite apparent contact angle. Strikingly, this contracted droplet, above a threshold temperature, well below the boiling point of the liquid, starts to spontaneously move on the substrate in an apparently erratic way. We describe and quantify this self-propulsion of the volatile drop. It arises due to spontaneous symmetry breaking of thermal Marangoni convection, which is induced by the nonuniform evaporation of the droplet. Using infrared imaging, we reveal the characteristic interfacial flow patterns associated with Marangoni convection in the evaporating drop. A scaling relation describes the correlation between the moving velocity of the drop and the apparent contact angle, both of which increase with the substrate temperature.