The newly discovered 39-K superconductor MgB2 holds great promise for superconducting electronics. Like the conventional superconductor Nb, MgB2 is a phonon-mediated superconductor, with a relatively long coherence length. These properties make the prospect of fabricating reproducible uniform Josephson junctions, the fundamental element of superconducting circuits, much more favourable for MgB2 than for high-temperature superconductors. The higher transition temperature and larger energy gap of MgB2 promise higher operating temperatures and potentially higher speeds than Nb-based integrated circuits. However, success in MgB2 Josephson junctions has been limited because of the lack of an adequate thin-film technology. Because a superconducting integrated circuit uses a multilayer of superconducting, insulating and resistive films, an in situ process in which MgB2 is formed directly on the substrate is desirable. Here we show that this can be achieved by hybrid physical-chemical vapour deposition. The epitaxially grown MgB2 films show a high transition temperature and low resistivity, comparable to the best bulk samples, and their surfaces are smooth. This advance removes a major barrier for superconducting electronics using MgB2.