The results are reported of high-precision pulsed-field gradient (PFG) NMR measurements concerning the transport of protons along the surface of purple membrane (PM) at room temperature. PM stacks were studied as a function of orientation and hydration. At 100% relative humidity the following results were obtained : (1) Translational diffusion of protons on the PM surface is fast, although about five times slower than in bulk water at room temperature; D(s) = 4.4 x 10(-6) cm2/s. (2) Fast diffusion was observed only parallel, but not perpendicular, to the PM planes, i.e. this diffusion process is clearly two-dimensional on the timescale of our experiments. (3) The diffusion is much slower at 86% relative humidity; this observation may be related to the slowing-down of biological function of PM (and in particular the proton pumping mechanism of bacteriorhodopsin) with decreasing hydration level. These results are in good agreement with those of a preliminary analysis of quasielastic incoherent neutron scattering (QINS) data. A new, improved analysis of the latter, making use of the NMR value of D(s) and including multiple scattering (MSC) corrections, is also presented. Finally the relation of the proton diffusion results with proton conductivity is discussed.