Unidirectional solidification of a dilute alloy (tin-bismuth) in a two-dimensional rectangular cavity is investigated using a uniform computational grid. The energy equation is served for the temperature held, while the species equation is solved for the solute distribution. A vorticity-vector potential representation is used for the governing equations for the velocity field. The constitutive equations are solved using a true transient method. An Alternating Direction Implicit (ADI) approximation is employed in the solution scheme for the vorticity while the conjugate gradient method is used for the vector potential equation. The results obtained from the numerical simulations compare very well with experimental and numerical results for directional solidification in the literature in terms of the propagation of the solidification front as well as the free convection flow patterns in the liquid.