Typical experiments to measure permeability in biologically relevant cell membranes utilize vesicles, single bilayer membrane capsules, as idealized cells. The permeation of solute through the vesicle membrane is analyzed with the assumption that the vesicle volume and membrane area are fixed during the experiment. In response to the excess solute concentration internal to the vesicle, however, an osmotic stress is set up that acts to stretch the vesicle membrane, resulting in an initial dilution of the vesicle contents, followed by a continued elastic response of the vesicle membrane as the experiment proceeds and the internal and external concentrations equilibrate. We derive a simple analytic theory of this process that couples the mass‐transfer process with the area elasticity of the vesicle membrane. If dilution of the exterior solution is used to initiate the permeation, significant effects on measured permeabilities are predicted for membranes with a low elastic modulus, when large concentration differences are imposed or vesicles are greater than 0.1 micron in diameter. In practice, this leads to an underestimate of the permeability of membranes composed of polyunsaturated lipids or lipids in contact with alcohols or alkanes. It could also lead to the erroneous conclusion that the permeability is concentration‐dependent. The effects of elasticity, however, are minor, if dialysis is used to continually remove any exterior solute or if saturated phospholipid membranes with a large elastic modulus or vesicles of less than 0.1 micron diameter are used.