We analyze the first two years of data from the Fermi Gamma Ray Space Telescope from the direction of the inner 10 degrees around the Galactic Center with the intention of constraining, or finding evidence of, annihilating dark matter. We find that the morphology and spectrum of the emission between 1.25 degrees and 10 degrees from the Galactic Center is well described by a the processes of decaying pions produced in cosmic ray collisions with gas, and the inverse Compton scattering of cosmic ray electrons in both the disk and bulge of the Inner Galaxy, along with gamma rays from known points sources in the region. The observed spectrum and morphology of the emission within approximately 1.25 degrees (~175 parsecs) of the Galactic Center, in contrast, departs from the expectations for by these processes. Instead, we find an additional component of gamma ray emission that is highly concentrated around the Galactic Center. The observed morphology of this component is consistent with that predicted from annihilating dark matter with a cusped (and possibly adiabatically contracted) halo distribution (density proportional to r^{-gamma}, with gamma=1.18 to 1.33). The observed spectrum of this component, which peaks at energies between 1-4 GeV (in E^2 units), can be well fit by a 7-10 GeV dark matter particle annihilating primarily to tau leptons with a cross section in the range of 4.6 x 10^-27 to 5.3 x 10^-26 cm^3/s, depending on how the dark matter distribution is normalized. We also discuss other sources for this emission, including the possibility that much of it originates from the Milky Way's supermassive black hole.

23 pages, 16 figures