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 $geq$ 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 $1^circ$ of previously detected TeV emitters, and twenty sources that are more than $1^circ$ away from any previously detected TeV source. Of these twenty new sources, fourteen have a potential counterpart in the fourth textit{Fermi} Large Area Telescope catalog of gamma-ray sources. We also explore potential associations of 3HWC sources with pulsars in the ATNF pulsar catalog and supernova remnants in the Galactic supernova remnant catalog.
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