The two-dimensional steady-state shape of a solidified region, such as a frost layer, was determined analytically for formation on a plate that is convectively cooled. The nonuniform shape of the layer is produced by exposure to a spatially nonuniform distribution of radiant energy. For high convective cooling the cooled wall approaches a uniform temperature, and an exact solution is obtained for the free boundary shape. For a lesser amount of convective cooling, the variation temperature along the cooled boundary is treated by a boundary perturbation method. Some illustrative examples are given that show the effects of nonuniform heating and the magnitude of convective heat transfer at the cooled wall. Only one boundary condition is approximated by the perturbation solution; all of the other boundary conditions are satisfied exactly. The calculated results given here were found to satisfy the approximate boundary condition within a very small error.