Recent high-pressure studies of condensed matter at extreme densities have uncovered various new phenomena in simple molecular and elemental substances(1). One of the most significant pressure-induced changes in materials properties is the transformation of insulators into metals and superconductors. Previous studies of compressed sulphur indicated transitions to metallic phases at 90 GPa (ref. 2) and 162 GPa (ref. 3). Here we demonstrate that at 93 GPa, elemental sulphur transforms not only to a metal, but also to a superconductor with a transition temperature, T-c, of 10.1 K. Using a highly sensitive magnetic susceptibility technique adapted for megabar-pressure diamond anvil cells, we find that T-c increases linearly with pressure up to 157 GPa. This contrasts with the negative dT(c)/dP observed(4) (at much lower pressures) in the heavier superconducting chalcogenides Se and Te. Moreover, at the transformation in sulphur to a higher pressure metallic phase near 160 GPa, T-c increases from 14 to 17 K. These are the highest reported transition temperatures for an elemental solid. As such, these results may provide crucial tests of mechanisms of superconductivity.