A solid is bounded by two concentric spherical surfaces and the whole system is at a temperature TA which is below the freezing point Tf of the substance. At a certain instant, one of the surfaces is raised to and maintained at a temperature TR which is above the freezing point TR which the other surface is maintained at the original temperature TA. The problem of finding the temperature distribution in the solid and the liquid phases and the position of the interface between the two phases and the rate of motion of the interface are solved by an analytic method in which a solution of the heat conduction problem is constructed by the superposition of solutions such that all the initial, asymptotic and boundary conditions of the problem are satisfied.