We present a new catalog of TeV gamma-ray sources using 1523 days of data from the High-Altitude Water Cherenkov (HAWC) Observatory. The catalog represents the most sensitive survey of the northern gamma-ray sky at energies above several TeV, with three times the exposure compared to the previous HAWC catalog, 2HWC. We report 65 sources detected at >=5{sigma} significance, along with the positions and spectral fits for each source. The catalog contains eight sources that have no counterpart in the 2HWC catalog, but are within 1{deg} of previously detected TeV emitters, and 20 sources that are more than 1{deg} away from any previously detected TeV source. Of these 20 new sources, 14 have a potential counterpart in the fourth Fermi Large Area Telescope catalog of gamma-ray sources. We also explore potential associations of 3HWC sources with pulsars in the Australia Telescope National Facility (ATNF) pulsar catalog and supernova remnants in the Galactic supernova remnant catalog.