In this paper, the control problem associated with a diffusion system with a free boundary is considered. This problem arises physically from attempting to control the rate of solidification of a liquid. An estimate for the reachable set of free boundary points corresponding to a specified class of control functions is derived. The control problem is first formulated as an infinite-dimensional mathematical programming problem, and then approximated by a corresponding finite-dimensional problem, The convergence of the approximation is established. Typical numerical solutions are also presented.