A finite difference scheme is developed to treat the solidification of an alloy system in a semiinfinite domain with prescribed surface temperatures. Constant properties are assumed in each region but may be different from one region to the other. The solid fraction distribution within the freezing zone is assumed to be a linear function of distance. Furthermore, the density variations in the mushy region are taken into consideration. A limiting solution for freezing rate is found to agree well with the existing solution in the literature.