The third catalog of high-energy gamma-ray sources detected by the EGRET telescope on the Compton Gamma Ray Observatory 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). The 271 sources (E greater than 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. A figure is presented that gives approximate upper limits for gamma-ray sources at any point in the sky, as well as information about sources listed in the second catalog and its supplement which do not appear in this catalog