A long-standing paradigm in astrophysics is that collisionsor mergers-of two neutron stars form highly relativistic and collimated outflows (jets) that power gamma-ray bursts of short (less than two seconds) duration(1-3). The observational support for this model, however, is only indirect(4,5). A hitherto outstanding prediction is that gravitational-wave events from such mergers should be associated with gamma-ray bursts, and that a majority of these bursts should be seen off-axis, that is, they should point away from Earth(6,7). Here we report the discovery observations of the X-ray counterpart associated with the gravitational-wave event GW170817. Although the electromagnetic counterpart at optical and infrared frequencies is dominated by the radioactive glow (known as a 'kilonova') from freshly synthesized rapid neutron capture (r-process) material in the merger ejecta(8-10), observations at X-ray and, later, radio frequencies are consistent with a short gamma-ray burst viewed off-axis(7,11). Our detection of X-ray emission at a location coincident with the kilonova transient provides the missing observational link between short gamma-ray bursts and gravitational waves from neutron-star mergers, and gives independent confirmation of the collimated nature of the gamma-ray-burst emission.