An anomalous diffusion version of a limit Stefan melting problem is posed. In this problem, the governing equation includes a fractional time derivative of order 0 < beta <= 1 and a fractional space derivative for the flux of order 0 < alpha <= 1. Solution of this fractional Stefan problem predicts that the melt front advance as s = t(gamma), gamma = beta/alpha+1. This result is consistent with fractional diffusion theory and through appropriate choice of the order of the time and space derivatives, is able to recover both sub-diffusion and super-diffusion behaviors for the melt front advance. (C) 2010 Elsevier Ltd. All rights reserved.