A simple batch sorption method is examined for diffusivity determinations of binary solutions of low molecular weight organic compounds in amorphous polymers. The method is based on solute uptake from a stirred liquid solution by a polymer sheet sample submerged in the liquid solution and measurement of the liquid-phase concentration as a function of time. A transport model is discussed for the transient process of mass transfer between the stirred solution and the polymer phase, accounting for swelling of the polymer sheet and for concentration dependence of the diffusivity. The diffusion coefficient is evaluated by curve fitting of the results of numerical solution of the model equations to experimental data. Experimental toluene diffusivity data for the elastomers EPDM and PDMS are presented, as determined by the batch sorption method using an aqueous solution as the liquid phase. Good agreement between the experimental data and the calculated sorption curves is observed. A key parameter of a free-volume theory for diffusion in amorphous polymer was determined, by which the binary diffusion coefficient can be determined as a function of the composition and temperature, far a variety of diffusing solute compounds in the two elastomers examined.