  The source-count distribution as a function of their flux, dN/dS, is one of
the main quantities characterizing gamma-ray source populations. We employ
statistical properties of the Fermi Large Area Telescope (LAT) photon counts
map to measure the composition of the extragalactic gamma-ray sky at high
latitudes (|b|>30 deg) between 1 GeV and 10 GeV. We present a new method,
generalizing the use of standard pixel-count statistics, to decompose the total
observed gamma-ray emission into (a) point-source contributions, (b) the
Galactic foreground contribution, and (c) a truly diffuse isotropic background
contribution. Using the 6-year Fermi-LAT data set (P7REP), we show that the
dN/dS distribution in the regime of so far undetected point sources can be
consistently described with a power law of index between 1.9 and 2.0. We
measure dN/dS down to an integral flux of ~2x10^{-11} cm^{-2}s^{-1}, improving
beyond the 3FGL catalog detection limit by about one order of magnitude. The
overall dN/dS distribution is consistent with a broken power law, with a break
at 2.1^{+1.0}_{-1.3}x10^{-8} cm^{-2}s^{-1}. The power-law index
n_1=3.1^{+0.7}_{-0.5} for bright sources above the break hardens to
n_2=1.97+-0.03 for fainter sources below the break. A possible second break of
the dN/dS distribution is constrained to be at fluxes below 6.4x10^{-11}
cm^{-2}s^{-1} at 95% confidence level. The high-latitude gamma-ray sky between
1 GeV and 10 GeV is shown to be composed of ~25% point sources, ~69.3% diffuse
Galactic foreground emission, and ~6% isotropic diffuse background.
