The Chandra COSMOS Survey (C-COSMOS) is a large, 1.8Ms, Chandra program that has imaged the central 0.5deg^2^ of the COSMOS field (centered at 10h, +02d) with an effective exposure of ~160ks, and an outer 0.4deg^2^ area with an effective exposure of ~80ks. The limiting source detection depths are 1.9x10^-16^erg/cm2/s in the soft (0.5-2keV) band, 7.3x10^-16^erg/cm2/s in the hard (2-10keV) band, and 5.7x10^-16^erg/cm2/s in the full (0.5-10keV) band. Here we describe the strategy, design, and execution of the C-COSMOS survey, and present the catalog of 1761 point sources detected at a probability of being spurious of <2x10^-5^ (1655 in the full, 1340 in the soft, and 1017 in the hard bands). By using a grid of 36 heavily (~50%) overlapping pointing positions with the ACIS-I imager, a remarkably uniform (+/-12%) exposure across the inner 0.5deg^2^ field was obtained, leading to a sharply defined lower flux limit. The widely different point-spread functions obtained in each exposure at each point in the field required a novel source detection method, because of the overlapping tiling strategy, which is described in a companion paper. This method produced reliable sources down to a 7-12 counts, as verified by the resulting logN-logS curve, with subarcsecond positions, enabling optical and infrared identifications of virtually all sources, as reported in a second companion paper.