  Submillimetre surveys during the past decade have discovered a population of
luminous, high-redshift, dusty starburst galaxies. In the redshift range 1 <= z
<= 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 microns in the rest frame. At
1 <= z <= 4, the peak is redshifted to wavelengths between 200 and 500 microns.
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 microns. Combining our results at 500
microns with those at 24 microns, we determine that all of the FIRB comes from
individual galaxies, with galaxies at z >= 1.2 accounting for 70 per cent of
it. As expected, at the longest wavelengths the signal is dominated by
ultraluminous galaxies at z > 1.
