  Past studies have identified a spatially extended excess of $\sim$1-3 GeV
gamma rays from the region surrounding the Galactic Center, consistent with the
emission expected from annihilating dark matter. We revisit and scrutinize this
signal with the intention of further constraining its characteristics and
origin. By applying cuts to the \textit{Fermi} event parameter CTBCORE, we
suppress the tails of the point spread function and generate high resolution
gamma-ray maps, enabling us to more easily separate the various gamma-ray
components. Within these maps, we find the GeV excess to be robust and highly
statistically significant, with a spectrum, angular distribution, and overall
normalization that is in good agreement with that predicted by simple
annihilating dark matter models. For example, the signal is very well fit by a
36-51 GeV dark matter particle annihilating to $b\bar{b}$ with an annihilation
cross section of $\sigma v = (1-3)\times 10^{-26}$ cm$^3$/s (normalized to a
local dark matter density of 0.4 GeV/cm$^3$). Furthermore, we confirm that the
angular distribution of the excess is approximately spherically symmetric and
centered around the dynamical center of the Milky Way (within
$\sim$$0.05^{\circ}$ of Sgr A$^*$), showing no sign of elongation along the
Galactic Plane. The signal is observed to extend to at least $\simeq10^{\circ}$
from the Galactic Center, disfavoring the possibility that this emission
originates from millisecond pulsars.
