  We have placed limits on the cosmological significance of gas-rich low
surface-brightness (LSB) galaxies as a proportion of the total population of
gas-rich galaxies by carrying out a very deep survey (HIDEEP) for neutral
hydrogen (HI) with the Parkes multibeam system. Such a survey avoids the
surface-brightness selection effects that limit the usefulness of optical
surveys for finding LSB galaxies. To complement the HIDEEP survey we have
digitally stacked eight 1-hour R-band Tech Pan films from the UK Schmidt
Telescope covering 36 square degrees of the survey area to reach a very deep
isophotal limit of 26.5 R mag/sq. arcsec. At this level, we find that all of
the 129 HI sources within this area have optical counterparts and that 107 of
them can be identified with individual galaxies. We have used the properties of
the galaxies identified as the optical counterparts of the HI sources to
estimate the significance of LSB galaxies (defined to be those at least 1.5
magnitudes dimmer in effective surface-brightness than the peak in the observed
distribution seen in optical surveys). We calculate the contribution of LSB
galaxies to the total number, neutral hydrogen density, luminosity density,
baryonic mass density, dynamical mass density and cross-sectional area of
gas-rich galaxies. We do not find any `Crouching Giant' LSB galaxies such as
Malin 1, nor do we find a population of extremely low surface-brightness
galaxies not previously found by optical surveys. Such objects must either be
rare, gas-poor or outside the survey detection limits.
