Sources of high-energy (>20 keV) bursts fall into two distinct types : the non-repeating gamma-ray bursters(1), several thousand of which have been detected but whose origin remains unknown, and the soft gamma-ray repeaters (SGRs), of which there are only three(2). The SGRs are known to be associated with supernova remnants, suggesting that the burst events most probably originate from young neutron stars(3). Here we report the detection of a third type of transient high-energy source. On 2 December 1995, we observed the onset of a sequence of hard X-ray bursts from a direction close to that of the Galactic Centre(4). The interval between bursts was initially several minutes, but after two days, the burst rate had dropped to about one per hour and has been largely unchanged since then. More than 1,000 bursts have now been detected, with remarkably similar light curves and intensities; this behaviour is unprecendented among transient X-ray and gamma-ray sources. We suggest that the origin of these bursts might be related to the spasmodic accretion of material onto a neutron star.