The existence of a boundary layer of different impurity concentration on the solid-melt interface during crystal growth into impure unstirred melts has been inferred by previous authors. The problem of how rapidly this layer attains its final form in the case of growth into an infinite unstirred melt, hitherto treated approximately, is shown to be capable of exact solution for all values of the segregation constant. The impurity distribution in the resultant solid is calculated and graphs are given for various values of the segregation constant.