Ion current densities near 1 A cm-2 at modest bias voltages (<200 mV) are reported for proton and deuteron transmission across single-layer graphene in polyelectrolyte-membrane (PEM)-style hydrogen pump cells. The graphene is sandwiched between two Nafion membranes and covers the entire area between two platinum-carbon electrodes, such that proton transfer is forced to occur through the graphene layer. Raman spectroscopy confirms that buried graphene layers are single-layer and relatively free of defects following the hot-press procedure used to make the sandwich structures. Area-normalized ion conductance values of approximately 29 and 2.1 S cm-2 are obtained for proton and deuteron transport, respectively, through single-layer graphene, following correction for contributions to series resistance from Nafion resistance, contact resistance, etc. These ion conductance values are several hundred to several thousand times larger than in previous reports on similar phenomena. A ratio of proton to deuteron conductance of 14 to 1 is obtained, in good agreement with but slightly larger than those in prior reports on related cells. Potassium ion transfer rates were also measured and are attenuated by a factor of many thousands by graphene, whereas proton transfer is attenuated by graphene by only a small amount. Rates for hydrogen and deuterium ion exchange across graphene were analyzed using a model whereby each hexagonal graphene hollow site is assumed to transmit ions with a specific per-site ion-transfer self-exchange rate constant. Rate constant values of approximately 2500 s-1 for proton transfer and 180 s-1 for deuteron transfer per site through graphene are reported.