  This is the second of two papers describing a model of galactic evolution in
the submillimetre waveband. The model incorporates a self-consistent treatment
of the evolution of dust and stars, is normalized to the submillimetre
properties of galaxies in the local universe, and can be used to make
predictions for both disk and elliptical galaxies and for `closed-box',
`inflow', and `outflow' models of galactic evolution. In Paper I we
investigated whether it is possible to explain the extreme dust masses of
high-redshift quasars and radio galaxies by galactic evolution. In this paper
we use the model to make predictions of the submillimetre background and source
counts. All our disk-galaxy models exceed at short wavelengths the
submillimetre background recently measured by Puget et al. (1996). We also find
that it is possible to fit the background over the entire wavelength range with
a elliptical model but not with a disk model. We make source count predictions
at 190 $\mu$m for the ISOPHOT instrument on ISO and at 850 $\mu$m for SCUBA. We
show that the shape of the 850 $\mu$ m source counts depends almost entirely on
the mass spectrum of the radiating objects. Finally, we consider the
limitations of the models. We find that one of the biggest uncertainties in the
model is our lack of information about the submillimetre properties of nearby
galaxies, in particular the lack of a direct measurement of the submillimetre
luminosity function.
