We have carried out an extremely long integration time (9000 s beam⁻¹) 21-cm blind survey of 60 deg² in Centaurus using the Parkes multibeam system. We find that the noise continues to fall as √tobs throughout, enabling us to reach an HI column-density limit of 4.2 × 10¹⁸ cm⁻² for galaxies with a velocity width of 200 km s⁻¹ in the central 32 deg² region, making this the deepest survey to date in terms of column density sensitivity. The HI data are complemented by very deep optical observations from digital stacking of multi-exposure UK Schmidt Telescope R-band films, which reach an isophotal level of 26.5 R mag arcsec⁻² (≃27.5 B mag arcsec⁻²). 173 HI sources have been found, 96 of which have been uniquely identified with optical counterparts in the overlap area. There is not a single source without an optical counterpart. Although we have not measured the column densities directly, we have inferred them from the optical sizes of their counterparts. All appear to have a column density of NH I= 10²⁰·⁶⁵±⁰·³⁸ . This is at least an order of magnitude above our sensitivity limit, with a scatter only marginally larger than the errors on NH I. This needs explaining. If confirmed it means that H I surveys will only find low surface brightness (LSB) galaxies with high MH I/LB . Gas-rich LSB galaxies with lower H I mass to light ratios do not exist. The paucity of low column-density galaxies also implies that no significant population will be missed by the all-sky H I surveys being carried out at Parkes and Jodrell Bank.