  We present a catalog of sources detected above 10 GeV by the Fermi Large Area
Telescope (LAT) in the first 7 years of data using the Pass 8 event-level
analysis. This is the Third Catalog of Hard Fermi-LAT Sources (3FHL),
containing 1556 objects characterized in the 10 GeV - 2 TeV energy range. The
sensitivity and angular resolution are improved by factors of 3 and 2 relative
to the previous LAT catalog at the same energies (1FHL). The vast majority of
detected sources (79%) are associated with extragalactic counterparts at other
wavelengths, including 16 sources located at very high redshift ($z>2$). Eight
percent of the sources have Galactic counterparts and 13% are unassociated (or
associated with a source of unknown nature). The high-latitude sky and the
Galactic plane are observed with a flux sensitivity of 4.4 to 9.5$\times
10^{-11}$ ph cm$^{-2}$ s$^{-1}$, respectively (this is approximately 0.5% and
1% of the Crab Nebula flux above 10 GeV). The catalog includes 214 new
$\gamma$-ray sources. The substantial increase in the number of photons (more
than 4 times relative to 1FHL and 10 times to 2FHL) also allows us to measure
significant spectral curvature for 32 sources and find flux variability for 163
of them. Furthermore, we estimate that for the same flux limit of $10^{-12}$
erg cm$^{-2}$ s$^{-1}$, the energy range above 10 GeV has twice as many sources
as above 50 GeV, highlighting the importance, for future Cherenkov telescopes,
of lowering the energy threshold as much as possible.
