The work in this paper concerns a mathematical model of the contact melting process of a rectangular material in contact with a hot plate. The problem is described by a coupled system of heat equations in the solid and melt layer, fluid flow in the melt, a Stefan condition at the melt interface, and a force balance between the weight of the solid and the fluid pressure. Since the melt layer remains thin throughout the process, we use the lubrication approximation to the fluid equations and assume that the heat flow in the fluid is dominated by conduction across the thin film. In the solid we employ a heat balance integral method. Results show that the film height has initial and final rapid increases, whereas for intermediate times the height slowly increases. The quasisteady state of previous models is never attained: This is shown to be an effect of neglecting the change in mass and conduction in the solid. The previously observed initial infinite velocity of the melt is shown to be a result of the perfect thermal contact assumption. For a water-ice system the melting rate is shown to be approximately linear, this allows us to reduce the problem to solving a single first order differential equation for the liquid layer thickness. The main analysis is carried out in two dimensions, but we briefly highlight the extension to three dimensions. The method is verified by comparison with previously published experimental results on the melting of n-octadecane. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2990751]