We have performed the first systematic search of the full Galaxy Evolution Explorer (GALEX) data archive for astrophysical variability on timescales of seconds to minutes by rebinning data across the whole mission to 30s time resolution. The result is the GALEX Flare Catalog (GFCAT), which describes 1426 ultraviolet variable sources, including stellar flares, eclipsing binaries, {delta} Scuti and RR Lyrae variables, and active galactic nuclei. Many of these sources have never previously been identified as variable. We have also assembled a table of observations of ultraviolet flares and accompanying statistics and measurements, including energies, and of candidate eclipsing stars. This effort was enabled by a significantly enhanced version of the gPhoton software for analyzing time-domain GALEX data; this "gPhoton2" package is available to support follow-on efforts.