We construct empirical models of the diffuse gamma-ray background toward the Galactic Center. Including all known point sources and a template of emission associated with interactions of cosmic rays with molecular gas, we show that the extended emission observed previously in the Fermi Large Area Telescope data toward the Galactic Center is detected at high significance for all permutations of the diffuse model components. However, we find that the fluxes and spectra of the sources in our model change significantly depending on the background model. In particular, the spectrum of the central Sgr A* source is less steep than in previous works and the recovered spectrum of the extended emission has large systematic uncertainties, especially at lower energies. If the extended emission is interpreted to be due to dark matter annihilation, we find annihilation into pure b-quark and τ-lepton channels to be statistically equivalent goodness of fits. In the case of the pure b-quark channel, we find a dark matter mass of 39.4(+3.7−2.9 stat)(±7.9 sys) GeV, while a pure τ+τ−-channel case has an estimated dark matter mass of 9.43(+0.63−0.52stat)(±1.2 sys) GeV. Alternatively, if the extended emission is interpreted to be astrophysical in origin such as due to unresolved millisecond pulsars, we obtain strong bounds on dark matter annihilation, although systematic uncertainties due to the dependence on the background models are significant.