We report the discovery of the low-density, transiting giant planet WASP-31b. The planet is 0.48 Jupiter masses and 1.55 Jupiter radii. It is in a 3.4-day orbit around a metal-poor, late-F-type, V=11.7 dwarf star, which is a member of a common proper motion pair. In terms of its low density, WASP-31b is second only to WASP-17b, which is a more highly irradiated planet of similar mass.