  The dark matter halo of the Milky Way is predicted to contain a very large
number of smaller subhalos. As a result of the dark matter annihilations taking
place within such objects, the most nearby and massive subhalos could appear as
point-like or spatially extended gamma-ray sources, without observable
counterparts at other wavelengths. In this paper, we use the results of the
Aquarius simulation to predict the distribution of nearby subhalos, and compare
this to the characteristics of the unidentified gamma-ray sources observed by
the Fermi Gamma-Ray Space Telescope. Focusing on the brightest high latitude
sources, we use this comparison to derive limits on the dark matter
annihilation cross section. For dark matter particles lighter than ~200 GeV,
the resulting limits are the strongest obtained to date, being modestly more
stringent than those derived from observations of dwarf galaxies or the
Galactic Center. We also derive independent limits based on the lack of
unidentified gamma-ray sources with discernible spatial extension, but these
limits are a factor of ~2-10 weaker than those based on point-like subhalos.
Lastly, we note that four of the ten brightest high-latitude sources exhibit a
similar spectral shape, consistent with 30-60 GeV dark matter particles
annihilating to b quarks with an annihilation cross section on the order of
sigma v ~ (5-10) x 10^-27 cm^3/s, or 8-10 GeV dark matter particles
annihilating to taus with sigma v ~ (2.0-2.5) x 10^-27 cm^3/s.
