Most galactic nuclei are now believed to harbour supermassive black holes. Studies of stellar motions 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 million solar masses. This object is spatially coincident with Sagittarius A* (Sgr A*), the unique compact radio source located at the dynamical centre of our Galaxy. By analogy with distant quasars and nearby active galactic nuclei (AGN), Sgr A* is thought to be powered by the gravitational potential energy released by matter as it accretes onto a supermassive black hole. However, Sgr A* is much fainter than expected in all wavebands, especially in X-rays, casting some doubt on this model. Recently, we reported the first strong evidence of X-ray emission from Sgr A*. Here we report the discovery of rapid X-ray flaring from the direction of Sgr A*. These data provide compelling evidence that the X-ray emission is coming from accretion onto a supermassive black hole at the Galactic Centre, and the nature of the variations provides strong constraints on the astrophysical processes near the event horizon of the black hole.