A numerical method is developed to solve a problem of coupled solidification and natural convection within a two-dimensional rectangular cavity. The vertical walls of the cavity are adiabatic while the horizontal walls are isothermal, the upper boundary being maintained below the fusion temperature of the liquid. The solid layer that forms is assumed to have a uniform thickness. The Boussinesq system of equations is used to describe the liquid, and the one-dimensional heat conduction equation is used to describe the solid. The average temperature derivatives at the solid-liquid interface are used to determine its motion. A coordinate transformation based on solid thickness and an alternating direction implicit differencing scheme are the central features of the numerical method. For the limited range of non-dimensional parameters studied, steady state values of solid thickness and heat transfer coefficient as well as stream function and temperature distributions are obtained. It is found that the steady state of the liquid is not affected by the presence of a moving interface during the transient period. Internal natural convection without phase change is then studied to learn more about the details of the flow field.