A linear perturbation method is used to solve the two-dimensional heat conduction problem in which a liquid, initially at the melting temperature, becomes solidified by heat transfer to a plane mold the temperature of which is approximately uniform, but contains a small sinusoidal perturbation in one space dimension. Results are obtained for the consequent sinusoidal perturbation in the melt/solid boundary as a function of time and for the temperature distribution throughout the solid shell. These results are expressed in terms of a series of confluent hypergeometric functions which shows good convergence for practical values of material properties and temperatures. The inverse problem, in which the melt/solid boundary is prescribed and the mold temperature is to be determined, is also briefly discussed.