  The Fermi Large Area Telescope (LAT) Collaboration has recently released the
Third Catalog of Hard Fermi-LAT Sources (3FHL), which contains 1556 sources
detected above 10 GeV with seven years of Pass 8 data. We investigate the
source count distribution of 3FHL sources at Galactic latitudes
$|b|>20^{\circ}$, where the sources are mostly blazars. We use two
complementary techniques: 1) a source-detection efficiency correction method
and 2) an analysis of pixel photon count statistics with the 1-point
probability distribution function (1pPDF). With the first method, using
realistic Monte Carlo simulations of the $\gamma$-ray sky, we calculate the
efficiency of the LAT to detect point sources. This enables us to find the
intrinsic source count distribution at photon fluxes down to
$7.5\times10^{-12}$ ph cm$^{-2}$s$^{-1}$. With this method we detect a flux
break at $(3.5\pm0.4) \times 10^{-11}
\mathrm{ph}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$ with a significance of at least
$5.4 \sigma$. The power-law indexes of the source count distribution above and
below the break are $2.09\pm0.04$ and $1.07\pm0.27$, respectively. This result
is confirmed with the 1pPDF method, which has a sensitivity reach of
$\sim10^{-11}$ ph cm$^{-2}$s$^{-1}$. Integrating the derived source count
distribution above the sensitivity of our analysis, we find that $(42\pm8)\%$
of the extragalactic $\gamma$-ray background originates from blazars.
