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We study a nonlocal version of the one-phase Stefan problem which takes into account midrange interactions, a model of phase transition which may be of interest at a certain mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The presence of midrange interactions leads to new phenomena which are not present in the usual local version of the one-phase Stefan model, namely, the creation of mushy regions, the existence of waiting times during which the liquid region does not move, and the possibility of melting nucleation. If the kernel is suitably rescaled, the corresponding solutions converge to the solution of the local one-phase Stefan problem. We prove that the model is well posed and give several qualitative properties. In particular, the long-time behavior is identified by means of a nonlocal obstacle problem.