Many present and future applications of superconductivity would benefit from electrostatic control of carrier density and tunnelling rates, the hallmark of semiconductor devices. One particularly exciting application is the realization of topological superconductivity as a basis for quantum information processing. Proposals in this direction based on the proximity effect in semiconductor nanowires are appealing because the key ingredients are currently in hand. However, previous instances of proximitized semiconductors show significant tunnelling conductance below the superconducting gap, suggesting a continuum of subgap states--a situation that nullifies topological protection. Here, we report a hard superconducting gap induced by the proximity effect in a semiconductor, using epitaxial InAs-Al semiconductor-superconductor nanowires. The hard gap, together with favourable material properties and gate-tunability, makes this new hybrid system attractive for a number of applications, as well as fundamental studies of mesoscopic superconductivity.