Submillimetre surveys during the past decade have discovered a population of luminous, high-redshift, dusty starburst galaxies. In the redshift range 1 less than or equal to z less than or equal to 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 30K with spectral energy distributions peaking at 100um in the rest frame. At 1 less than or equal to z less than or equal to 4, the peak is redshifted to wavelengths between 200 and 500um. 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 500um. Combining our results at 500 mum with those at 24um, we determine that all of the FIRB comes from individual galaxies, with galaxies at z greater than or equal to 1.2 accounting for 70% of it. As expected, at the longest wavelengths the signal is dominated by ultraluminous galaxies at z>1.