  We use an 850-micron SCUBA map of the HDF to study the dust properties of
optically-selected starburst galaxies at high redshift. Although the 850-micron
map does not allow detection of the majority of individual sources, we show
that the galaxies with the highest UV star-formation rates are detected
statistically, with a flux density of about S_{850}=0.2 mJy for an apparent UV
star-formation rate of 1 h^{-2} msun yr^{-1}. This level of submillimetre
output indicates that the total star-forming activity is on average a factor of
approximately 6 times larger than the rate inferred from the UV output of these
galaxies. The general population of optical starbursts is then predicted to
contribute at least 25 per cent of the 850-micron background. We carry out a
power-spectrum analysis of the map, which yields some evidence for angular
clustering of the background source population, but at a level lower than that
seen in Lyman-break galaxies. Together with other lines of argument,
particularly from the NICMOS HDF data, this suggests that the 850-micron
background originates over an extremely wide range of redshifts -- perhaps 1 <
z < 6.
