We present a catalog of 203 clusters of galaxies serendipitously detected in 647 ROSAT PSPC high Galactic latitude pointings covering 158 deg². This is one of the largest X-ray-selected cluster samples, comparable in size only to the ROSAT All-Sky Survey sample of nearby clusters (Ebeling et al. 1997). We detect clusters in the inner 175 of the ROSAT PSPC field of view using the spatial extent of their X-ray emission. Fluxes of detected clusters range from 1.6 × 10^-14 to 8 × 10^-12 ergs s^-1 cm^-2 in the 0.5-2 keV energy band. X-ray luminosities range from 10⁴² ergs s^-1, corresponding to very poor groups, to ~5 × 10⁴⁴ ergs s^-1, corresponding to rich clusters. The cluster redshifts range from z = 0.015 to z > 0.5. The catalog lists X-ray fluxes, core radii, and spectroscopic redshifts for 73 clusters and photometric redshifts for the remainder. Our detection method, optimized for finding extended sources in the presence of source confusion, is described in detail. Selection effects necessary for a statistical analysis of the cluster sample are comprehensively studied by Monte Carlo simulations. We have optically confirmed 203 of 223 X-ray sources as clusters of galaxies. Of the remaining 20 sources, 19 are likely false detections arising from blends of unresolved point X-ray sources. Optical identifications of the remaining object are hampered by a bright nearby star. Above a flux of 2 × 10^-13 ergs s^-1 cm^-2, 98% of extended X-ray sources are optically confirmed clusters. The number of false detections and their flux distribution are in perfect agreement with simulations. The log N-log S relation for clusters derived from our catalog shows excellent agreement with counts of bright clusters derived from the Einstein Extended Medium Sensitivity Survey and ROSAT All-Sky Survey. At fainter fluxes, our log N-log S relation agrees with the smaller area WARPS survey. Our cluster counts appear to be systematically higher than those from a 50 deg² survey by Rosati et al. In particular, at a flux of 2 × 10^-13 ergs s^-1 cm^-2, we find a surface density of clusters of 0.57 ± 0.07 deg^-2, which is a factor of 1.3 more than was found by Rosati et al. This difference is marginally significant at the ~2 σ level. The large area of our survey makes it possible to study the evolution of the X-ray luminosity function in the high luminosity range inaccessible with other, smaller area ROSAT surveys.