A one-dimensional heat transfer problem in the phase-change slab, one side of which is isothermal while the other is insulated, is considered. Both cases—fusion and solidification—are treated. Slab temperature at the intitial moment is assumed constant and not critical. The main goal of this paper is to find the additional time required for a total phase change, compared with the case of the critical initial temperature. By analogy with perturbation problems in hydrodynamics, an appropriate solution is constructed consisting of an inner and an outer solution. The evaluation of the maximum error of the integral heat balance equation of the slab is treated as an indirect evaluation of the accuracy of the solution obtained. This evaluation shows that the solution can provide sufficient accuracy only in cases in which at least one of the three nondimensional parameters of the problem is small.