Heat transfer accompanied by phase change in a semitransparent medium subject to a time-variant surface temperature is investigated. A theoretical analysis is developed for the optically thick material. The Rosseland equation for local radiative flux is employed. Because of the inherent nonlinearity of the radiative diffusion term in the energy equation and the moving boundary condition involved in the problem, the exact solution is not feasible. An analytical approximation technique known as the heat balance integral method is, therefore, adopted in the present analysis. Solutions of the solidification (or melting) line are obtained for various time-dependent surface temperatures, e.g., surface temperature as a linear or an exponential function of time. The results based on the present study illustrate the effect of radiative transport on the solidification rate and on the temperature distribution in the semitransparent medium. Under certain limiting conditions, the present solutions are found to reduce to the results of some earlier analysis in the literature.