The results of an experimental study into the coupling of mass and momentum transfer for dilute polymer solutions are presented to test the numerous theoretical investigations of this effect. Solutions of high molecular weight, nearly monodisperse, linear polystyrene in oligomeric polystyrene were sheared in the cone-and-plate geometry. After long times under shear, the concentration of the high molecular weight species was measured as a function of radial position using a gel permeation chromatography based technique. An increase in polymer concentration of over 200% near the apex of the cone and a depletion of polymer near the edge was measured indicating a migration of polymer as a result of shearing the sample. The time evolution of the nonuniform concentration profile was found to be very slow and a strong function of both the polymer molecular weight and the applied shear rate. The experimental results are compared to theoretical predictions of a dumbbell based model and measured values of the physical parameters. While the method of solution of the highly coupled problem invokes a number of simplifying assumptions, the comparison to experimental results suggests that the theory substantially underestimates the rate of migration.