A “front-tracking” finite-element method has been developed for the study of inverse one-dimensional conduction heat transfer problems with phase change. The method can be used to determine the transient position of the change of phase interface and the thermal history in the solid region from information concerning the temperature history at two arbitrary points in the solid region. This technique can be employed in the control and experimental evaluation of practical heat transfer problems with phase change, such as welding. The accuracy of the method is illustrated through numerical experimentation and found to depend on a typical dimensionless Fourier number.