Whereas phase-change problems associated with heat conduction have been well studied during the last three decades, very little attention has been paid to phase changes taking place in convective heat diffusion. Numerical methods dealing with conventional phase change problems do not directly work in cases where a fluid is concerned. In this paper, an enthalpy method is extended to solve phase-change problems associated with fluids. By using the concept enthalpy, the governing equations are first reformulated into a single quasi-linear partial differential equation that implicitly takes into account the condition of phase change. This equation together with appropriate initial and boundary conditions are then decomposed into two sets of equations respectively representing a convection and a diffusion problem. The decomposition is accomplished in such a way that no phase contradiction occurs between the two separate problems. The convection problem is solved by the method of step by step characteristics and the diffusion problem by a Galerkin finite element method. Numerical examples demonstrate that the numerical method produces reasonable results.