Submillimetre surveys during the past decade have discovered a population of luminous, high-redshift, dusty starburst galaxies. In the redshift range 1 <or= z <or= 4, these massive submillimetre galaxies go through a phase characterized by optically obscured star formation at rates several hundred times that in the local Universe. Half of the starlight from this highly energetic process is absorbed and thermally re-radiated by clouds of dust at temperatures near 30 K with spectral energy distributions peaking at 100 microm in the rest frame. At 1 <or= z <or= 4, the peak is redshifted to wavelengths between 200 and 500 microm. The cumulative effect of these galaxies is to yield extragalactic optical and far-infrared backgrounds with approximately equal energy densities. Since the initial detection of the far-infrared background (FIRB), higher-resolution experiments have sought to decompose this integrated radiation into the contributions from individual galaxies. Here we report the results of an extragalactic survey at 250, 350 and 500 microm. Combining our results at 500 microm with those at 24 microm, we determine that all of the FIRB comes from individual galaxies, with galaxies at z >or= 1.2 accounting for 70% of it. As expected, at the longest wavelengths the signal is dominated by ultraluminous galaxies at z > 1.