The problem of solidification of two semi-infinite materials with arbitrarily prescribed initial conditions is studied. This is different from the classical Stefan problem; there are no prescribed boundary conditions. It is found that there are, depending on the prescribed initial conditions, four different possibilities: (i) solidification starts immediately and the interfacial boundary is of the order oft1/2 ast→0, (ii) solidification also starts immediately but the interfacial boundary is of an order other thant1/2, (iii) no solidification will ever occur, and (iv) there is a pre-solidification period during which the temperatures of both materials are redistributed and solidification will occur only after the surface temperature has reached the freezing point. Conditions for the occurrence of these cases are examined and established. The exact solutions for each of these cases are derived and discussed.