An accepted approach to computing laser-induced peak surface temperature is to employ the enthalpy formulation of the transient heat conduction equation [Grigoropulos et al, 1995; Iyer et al, 2017]. This approach is generally implemented using an explicit numerical scheme to solve the thermal transport equation. While it offers the advantage of modeling the solid-melt phase transition automatically, the approach results in instability-like behavior in the computed surface temperature. When laser-induced ablation becomes significant, the heating rate in the surface cell becomes unrealistically large. This results in spikes in the computed peak surface temperature due to large errors in calculating the heating rate. In this paper, we present a new approach, which we refer to as the Moving Frame Solver, that employs a moving coordinate frame of reference, located at the receding evaporating surface. We also use an analytical representation for the phase transition region of the enthalpy-temperature relationship. The Moving Frame Solver combined with an implicit scheme leads to a stable solution without surface temperature, pressure or velocity spikes. In other words, any instability in these computed parameters due to use of an explicit scheme has been eliminated. Details of the new thermal solver and example calculations are presented. Numerical experiments suggest that the surface cell size needs to be small, ~ 0.1 mm, to obtain a highly accurate solution with a typical metal such as aluminum. Using the Moving Frame Solver with a refined grid near the surface, but coarse elsewhere, enables accurate and stable surface temperature computation.