An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.