We study the effect of an interface barrier on the Tc of a superconducting-normal S-N metal sandwich in the Cooper limit. The calculation is based on the diffusion approximation of de Gennes and the barrier is accounted for by appropriate boundary conditions at the interface. In contrast with the original de Gennes result, it is shown that for finite thicknesses Tc does not depend only on the N to S thickness ratio but also on the absolute value of the thickness. An analysis of experimental data obtained on Pb-Cu sandwiches shows that this finite-thickness correction is important even at very small thicknesses and for rather transparent barriers. Assuming that the penetration probability shows only random variations in a series of samples prepared under the same nominal conditions, the best fit between theory and experiment is obtained for a BCS coupling parameter in Cu equal to 0.015.