  We show the existence of a statistically significant, robust detection of a
gamma-ray source in the Milky Way Galactic Center that is consistent with a
spatially extended signal using about 4 years of Fermi-LAT data. The gamma-ray
flux is consistent with annihilation of dark matter particles with a thermal
annihilation cross-section if the spatial distribution of dark matter particles
is similar to the predictions of dark matter only simulations. We find
statistically significant detections of an extended source with gamma-ray
spectrum that is consistent with dark matter particle masses of approximately
10 GeV to 1 TeV annihilating to b/b-bar quarks, and masses approximately 10 GeV
to 30 GeV annihilating to tau+ tau- leptons. However, a part of the allowed
region in this interpretation is in conflict with constraints from Fermi
observations of the Milky Way satellites. The biggest improvement over the fit
including just the point sources is obtained for a 30 GeV dark matter particle
annihilating to b/b-bar quarks. The gamma-ray intensity and spectrum are also
well fit with emission from a millisecond pulsar (MSP) population following a
density profile like that of low-mass X-ray binaries observed in M31. The
greatest goodness-of-fit of the extended emission is with spectra consistent
with known astrophysical sources like MSPs in globular clusters or cosmic ray
bremsstrahlung on molecular gas. Therefore, we conclude that the bulk of the
emission is likely from an unresolved or spatially extended astrophysical
source. However, the interesting possibility of all or part of the extended
emission being from dark matter annihilation cannot be excluded at present.
