At current densities large in magnitude and also large compared to the exchange current density, a conducting disk in an insulating plane has a very nonuniform current distribution across the surface. The current distribution near the center is governed predominantly by ohmic effects, but near the edge, electrode kinetics become important. This paper describes the treatment of the current and potential distributions on the electrode, especially near the edge of the electrode where the nonuniformity is most extreme. The electrode kinetics are taken to be in the Tafel region. The results are valid for any and all large values of current density and provide a definitive description of the way in which the current and potential distributions approach the purely ohmically controlled (primary) distributions as the current is increased. The current density at the edge of the disk is found to increase with the square of the average current density. These results for the edge region of a disk at high currents and those of an earlier paper for high exchange currents can be applied to electrodes of more general geometry.