  The ROSAT Deep Survey in the Lockman Hole is the most sensitive X-ray survey
performed to date, encompassing an exposure time of 207 ksec with the PSPC and
a total of 1.32 Msec with the HRI aboard ROSAT. Here we present the complete
catalogue of 50 X-ray sources with PSPC fluxes (0.5--2 keV) above $ 5.5 \times
10^{-15} erg cm^{-2} s^{-1}$. The optical identifications are discussed in an
accompanying paper (Schmidt et al., 1997). We also derive a new log(N)--log(S)
function reaching a source density of $970 \pm 150 deg^{-2}$ at a limiting flux
of $10^{-15} erg cm^{-2} s^{-1}$. At this level 70-80% of the 0.5--2 keV X-ray
background is resolved into discrete sources. Utilizing extensive simulations
of artificial PSPC and HRI fields we discuss in detail the effects of source
confusion and incompleteness both on source counts and on optical
identifications. Based on these simulations we set conservative limits on flux
and on off-axis angles, which guarantee a high reliability of the catalogue. We
also present simulations of shallower fields and show that surveys, which are
based on PSPC exposures longer than 50 ksec, become severely confusion limited
typically a factor of 2 above their $4\sigma$ detection threshold. This has
consequences for recent claims of a possible new source population emerging at
the faintest X-ray fluxes.
  Keywords: surveys -- cosmology: diffuse radiations -- X-rays: galaxies
