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 $4sigma$ 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
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