A source-and-sink method has been used to solve a Stefan problem imposed with a moving heal front travelling at constant velocity in a fixed direction. The problem is transformed to moving coordinates and Laplace transform is used to develop the exact solution of this problem in quasi-steady stale. Twelve cases have been studied that cover constant temperature and flux conditions imposed on the moving front. The interface positions and the temperatures in the medium can be derived in closed forms for eight cases. For the four cases whose solutions cannot be derived in a closed form, procedures for exact solution of the problems are given in great detail. Numerical examples are also provided for a parametric study of the problems.