The nuclei of most galaxies are now believed to harbour supermassive black holes. The motions of stars in the central few light years of our Milky Way Galaxy indicate the presence of a dark object with a mass of about 2.6 × 106 solar masses (refs 2, 3). This object is spatially coincident with the compact radio source Sagittarius A* (Sgr A*) at the dynamical centre of the Galaxy, and the radio emission is thought to be powered by the gravitational potential energy released by matter as it accretes onto a supermassive black hole. Sgr A* is, however, much fainter than expected at all wavelengths, especially in X-rays, which has cast some doubt on this model. The first strong evidence for X-ray emission was found only recently. Here we report the discovery of rapid X-ray flaring from the direction of Sgr A*, which, together with the previously reported steady X-ray emission, provides compelling evidence that the emission is coming from the accretion of gas onto a supermassive black hole at the Galactic Centre.