The boundary element technique is used to solve the steady state convection-diffusion problem with constant velocity in a two-dimensional domain with a free interface. These problems arise in a number of important heat transfer applications involving melting or solidification, such as bulk crystal growth in Bridgman furnaces. The boundary element approach reduces the dimension of the problem, thereby improving the computational efficiency, and is particularly well suited to free-surface problems in which the position and shape of the solid-liquid interface are of primary importance. Results are presented for a case study problem representing solidification in a two-dimensional, rectangular configuration.