We Present a mathematical model for the region of dendritic or cellular growth which often forms during the solidification of alloys. The model treats the region of mixed phase (solid and liquid) as a continuum whose properties vary with the local volume fraction of solid. It is assumed that transports of heat and of solute are by diffusion alone, and the model is closed by a condition of marginal equilibrium. Results are obtained for the unidirectional solidification of an alloy from a plane wall. The spatial variations of solid fraction are highly suggestive of the types of morphology that can occur, and a wealth of different structures are found as the physical parameters are varied. Although the model ignores gravity entirely, the results can be applied to the solidification from below of an alloy which is initially less dense than its eutectic. Predictions for the growth rate of the mixed-phase region agree well with existing experimental measurements of ice growing from aqueous salt solutions.