Coupling the surface state of a topological insulator to an s-wave superconductor is predicted to produce the long-sought Majorana quasiparticle excitations. However, superconductivity has not been measured in surface states when the bulk charge carriers are fully depleted, that is, in the true topological regime relevant for investigating Majorana modes. Here we report measurements of d.c. Josephson effects in topological insulator-superconductor junctions as the chemical potential is moved through the true topological regime characterized by the presence of only surface currents. We compare our results with three-dimensional quantum transport simulations, and determine the effects of bulk/surface mixing, disorder and magnetic field; in particular, we show that the supercurrent is largely carried by surface states, due to the inherent topology of the bands, and that it is robust against disorder. Our results thus clarify key open issues regarding the nature of supercurrents in topological insulators.