A fixed grid enthalpy method is developed for simulating the growth of a dendrtic crystal in an undercooled binary alloy melt held in a two-dimensional cavity. The approach solves the transport equations for mixture enthalpy and mixture concentration on a uniform grid of square control volumes. In dimensionless form the enthalpy is defined as the sum of the temperature and liquid fraction. The curvature of the solid-liquid interface is calculated by treating the liquid fraction in this definition as a "level set". Calculations of the curvature are then used, in computational cells on the interface, to set the temperature to the undercooled value obtained from the Gibbs-Thomson condition. Results are relatively free of grid anisotropy, approach the steady state tip velocity predicted from microscopic solubility theory, and predict realistic behavior for solute redistribution.