The temperature and the stress fields in the solidified layer and in the mold of finite thickness for a unidirectional casting process are investigated. Earlier solutions are extended to include the effect of the thermal capacity of the mold on the freezing front growth instability. A numerical solution is obtained for both the heat conduction and the residual stress problem. The results show that the perturbation in contact pressure tends asymptotically to a maximum value at larger times for the lower values of the thermal capacities of the mold materials. The magnitude of the contact pressure perturbation is decreased by the inclusion of the thermal capacity of the mold material, and this effect is enhanced for less distortive and thicker molds. The present article assumes that the thermal and mechanical problems are uncoupled along the casting mold interface. Despite this limitation, the results presented in this article indicate that a mold with a higher thermal capacity (or lower thermal diffusivity) might be less susceptible to thermoelastic instabilities associated with the contact pressure and its dependence on the thermal contact resistance at the casting mold interface.