A one-phase Stefan problem with a variable diffusivity is investigated. For two particular choices of diffusivity-one varying as a power law of position the other as a power function of the potential slope-exact similarity solutions are obtained. Unlike other similarity solutions that involve a time exponent n = 1/2, the derived solutions can exhibit exponents in the range 0 < n < 1. Application of these solutions in the verification of a numerical scheme highlights the importance of a correct numerical treatment for handling variations in diffusivity. 2012 Elsevier Ltd. All rights reserved.