The behavior of the one-phase Stefan problem describing the sorption of a finite amount of swelling solvent is studied. The solvent diffuses into the polymer, which causes it to change from a glassy to a rubbery state. The interface between these two states is characterized by a moving boundary, whose speed is defined by a kinetic law. A key distinctive feature of the model in this paper is that the polymer is exposed initially to a finite amount of penetrant and eventually approaches an equilibrium value. This equilibrium causes the time history of the interface to be more complex than that of the related, and more studied, problem where the swelling process occurs instantaneously at the penetration surface. We analyze the model using formal asymptotic expansions, for both small and large times as well as for the large control parameter. A numerical scheme is described which immobilizes the boundary and correctly identifies the appropriate starting solution. In addition, we discuss the heat balance integral method applied to this system and show that, while it does not give a particularly accurate approximation of the concentration, it is very good at capturing the behavior of the moving boundary position and also helps motivate our large time expansion. The form of the kinetic law is usually assumed to be linear, but this is a nonphysical restriction, and we are able to determine accurate asymptotic and numerical results for the nonlinear case.