A new data mode and new analysis methods are used to detect Terrestrial Gamma-ray Flashes (TGFs) with the Fermi Gamma-ray Burst Monitor (GBM) 10 times more frequently than previously. In 1037 h of observations at times and over regions for which TGFs are expected, 384 new TGFs were found in addition to the 39 TGFs and two Terrestrial Electron Beam events already detected without the new data mode and methodology. Cosmic ray showers were found to be an important background; they show characteristic signatures in the data of both GBM and the Fermi Large Area Telescope Calorimeter that enable their removal, leaving a sample estimated to consist of ≈98% TGFs. The sample includes shorter TGFs than previously found with GBM. The true duration distribution likely contains additional short TGFs because their detection by GBM is limited by detector dead time. One-third of this sample has matches with locations from the World Wide Lightning Location Network (WWLLN)—maps of these locations show the geographic and meteorological features more clearly than maps of spacecraft locations. The intrinsic TGF rate is evaluated using the lightning rate maps of the Lightning Imaging Sensor, accounting for the detection efficiency of GBM as a function of spacecraft-source offset, from which we estimate a global TGF rate of ≈400,000 per year. With continuous production of data in the new mode we estimate that GBM will detect ≈850 TGFs per year.