Hydrogen pump cells fabricated from Nafion membranes and Pt-on-carbon electrodes were used to measure rates of proton transfer across single-layer graphene embedded in Nafion | graphene | Nafion sandwich structures. Single-layer graphene prepared by chemical vapor deposition onto copper foil was sandwiched between Nafion membranes by a series of hot-pressing steps coupled with chemical etching to remove the copper. Proton current densities in excess of 1 A / cm² were observed for proton transfer across the graphene at cell bias voltages less than 200 mV. Confocal Raman spectroscopy measurements of the Nafion / graphene sandwich structures are consistent with single-layer graphene having very few structural defects embedded between Nafion membranes. Ion-transfer rates for lithium, sodium, potassium, and other cations were also measured and found to be much lower than rates for proton transfer, in most cases by many orders of magnitude. Two-layer and few-layer (3-5 layer) graphene sandwich structures were also studied and found to give proton currents much lower than single-layer graphene. Proton currents are described in terms of an area-normalized resistance following correction for all other contributions to series resistance in the cells. Resistances are then analyzed in terms of a rate constant for proton self-exchange across graphene.