We report on the observation of the bright, long gamma-ray burst, GRB 090926A, by the Gamma-ray Burst Monitor and Large Area Telescope (LAT) instruments on board the Fermi Gamma-ray Space Telescope. GRB 090926A shares several features with other bright LAT bursts. In particular, it clearly shows a short spike in the light curve that is present in all detectors that see the burst, and this in turn suggests that there is a common region of emission across the entire Fermi energy range. In addition, while a separate high-energy power-law component has already been observed in other gamma-ray bursts, here we report for the first time the detection with good significance of a high-energy spectral break (or cutoff) in this power-law component around 1.4 GeV in the time-integrated spectrum. If the spectral break is caused by opacity to electron-positron pair production within the source, then this observation allows us to compute the bulk Lorentz factor for the outflow, rather than a lower limit.