Fast radio bursts (FRBs) are brief, bright, extragalactic radio flashes. Their physical origin remains unknown, but dozens of possible models have been postulated. Some FRB sources exhibit repeat bursts. Though over a hundred FRB sources have been discovered to date, only four have been localised and associated with a host galaxy, with just one of the four known to repeat. The properties of the host galaxies, and the local environments of FRBs, provide important clues about their physical origins. However, the first known repeating FRB has been localised to a low-metallicity, irregular dwarf galaxy, and the apparently non-repeating sources to higher-metallicity, massive elliptical or star-forming galaxies, suggesting that perhaps the repeating and apparently non-repeating sources could have distinct physical origins. Here we report the precise localisation of a second repeating FRB source, FRB 180916.J0158+65, to a star-forming region in a nearby (redshift $z = 0.0337 \pm 0.0002$) massive spiral galaxy, whose properties and proximity distinguish it from all known hosts. The lack of both a comparably luminous persistent radio counterpart and a high Faraday rotation measure further distinguish the local environment of FRB 180916.J0158+65 from that of the one previously localised repeating FRB source, FRB 121102. This demonstrates that repeating FRBs have a wide range of luminosities, and originate from diverse host galaxies and local environments.