  The one-point function (i.e., the isotropic flux distribution) is a
complementary method to (anisotropic) two-point correlations in searches for a
gamma-ray dark matter annihilation signature. Using analytical models of
structure formation and dark matter halo properties, we compute the gamma-ray
flux distribution due to annihilations in extragalactic dark matter halos, as
it would be observed by the Fermi Large Area Telescope. Combining the central
limit theorem and Monte Carlo sampling, we show that the flux distribution
takes the form of a narrow Gaussian of `diffuse' light, with an `unresolved
point source' power-law tail as a result of bright halos. We argue that this
background due to dark matter constitutes an irreducible and significant
background component for point-source annihilation searches with galaxy
clusters and dwarf spheroidal galaxies, modifying the predicted signal-to-noise
ratio. A study of astrophysical backgrounds to this signal reveals that the
shape of the total gamma-ray flux distribution is very sensitive to the
contribution of a dark matter component, allowing us to forecast promising
one-point upper limits on the annihilation cross section. We show that by using
the flux distribution at only one energy bin, one can probe the canonical cross
section required for explaining the relic density, for dark matter of masses
around tens of GeV.
