This paper treats a one dimensional phase-change problem, 'ice melting', by a vertex-centered finite volume method. Numerical solutions are obtained by using two approaches where the first one is based on the heat conduction equation with the fixed grid, latent heat source approach (LHA), while the second uses the equivalent thermodynamic parameters defined by considering the apparent heat capacity method (AHC). A comparison between the two approaches is presented, furthermore the accuracy and flexibility of the numerical methods are verified by comparing the results with existing analytical solutions. Results indicate that one phase-change problems can be handled easily with excellent accuracies by using the AHC method.