In this paper we study a free boundary problem arising from a stress-driven diffusion in polymers. The main feature of the problem is that the mass flux of the penetrant is proportional to the gradient of the concentration and the gradient of the stress. A Maxwell-like viscoelastic relationship is assumed between the stress and the concentration. The phase change takes place on the interface between the glassy and rubbery states of the polymer and a Stefan-type of free boundary condition is imposed on the free boundary. It is shown that under certain conditions the problem has a unique weak solution.