In crystal-growth systems of the Bridgman-Stockbarger type, the solid-liquid interface touches a conducting wall which is the inner wall of the crucible. When the thermal conductivities of the solid and the liquid phases are different, which they usually are, the gradient of the axial thermal profile changes abruptly at the s - 1 interface. Since the thermal profile in the adjacent crucible cannot experience such a jump, the s - 1 interface must accommodate for this difference in thermal behavior at the wall. Using a fairly simple model example, we show that the s - 1 interface becomes singular there. The singularity is a weak logarithmic one.