We present a catalog of gamma-ray sources at energies above 10 GeV based on data from the Large Area Telescope (LAT) accumulated during the first 3 years of the Fermi Gamma-ray Space Telescope mission. This catalog complements the Second Fermi-LAT Catalog, which was based on 2 years of data extending down to 100 MeV and so included many sources with softer spectra below 10 GeV. The First Fermi-LAT Catalog of >10 GeV sources (1FHL) has 514 sources, and includes their locations, spectra, a measure of their variability, and associations with cataloged sources at other wavelengths. We found that 449 (87%) could be associated with known sources, of which 393 (76% of the 1FHL sources) are active galactic nuclei. We also highlight the subset of the 1FHL sources that are the best candidates for detection at energies above 50 GeV with ground-based gamma-ray observatories.