Asteroids are primitive Solar System bodies that evolve both collisionally and through disruptions arising from rapid rotation(1). These processes can lead to the formation of binary asteroids(2-4) and to the release of dust(5), both directly and, in some cases, through uncovering frozen volatiles. In a subset of the asteroids called main-belt comets, the sublimation of excavated volatiles causes transient comet-like activity(6-8). Torques exerted by sublimation measurably influence the spin rates of active comets(9) and might lead to the splitting of bilobate comet nuclei(10). The kilometre-sized main-belt asteroid 288P (300163) showed activity for several months around its perihelion 2011 (ref. 11), suspected to be sustained by the sublimation of water ice(12) and supported by rapid rotation(13), while at least one component rotates slowly with a period of 16 hours (ref. 14). The object 288P is part of a young family of at least 11 asteroids that formed from a precursor about 10 kilometres in diameter during a shattering collision 7.5 million years ago(15). Here we report that 288P is a binary main-belt comet. It is different from the known asteroid binaries in its combination of wide separation, near-equal component size, high eccentricity and comet-like activity. The observations also provide strong support for sublimation as the driver of activity in 288P and show that sublimation torques may play an important part in binary orbit evolution.