The third catalog of high-energy gamma-ray sources detected by the EGRET telescope on the Compton Gamma Ray Observatory (3EG) includes data from 1991 April 22 to 1995 October 3 (cycles 1, 2, 3, and 4 of the mission). In addition to including more data than the second EGRET catalog and its supplement, this catalog uses completely reprocessed data to correct a number of mostly minimal errors and problems. 14 sources (3 from the Supplement) listed in Table 6 below are not appearing in the 3EG. The 271 sources (E>100 MeV) in the catalog include the single 1991 solar flare bright enough to be detected as a source, the Large Magellanic Cloud, five pulsars, one probable radio galaxy detection (Cen A), and 66 high-confidence identifications of blazars (BL Lac objects, flat-spectrum radio quasars, or unidentified flat-spectrum radio sources). In addition, 27 lower confidence potential blazar identifications are noted. Finally, the catalog contains 170 sources not yet identified firmly with known objects, although potential identifications have been suggested for a number of those. This catalog supersedes the 2EG (Cat. <J/ApJS/101/259>) and its supplement (Cat. <J/ApJS/107/227>).