  The extended excess toward the Galactic Center (GC) in gamma rays inferred
from Fermi-LAT observations has been interpreted as being due to dark matter
(DM) annihilation. Here, we perform new likelihood analyses of the GC and show
that, when including templates for the stellar galactic and nuclear bulges, the
GC shows no significant detection of a DM annihilation template, even after
generous variations in the Galactic diffuse emission models and a wide range of
DM halo profiles. We include Galactic diffuse emission models with combinations
of three-dimensional inverse Compton maps, variations of interstellar gas maps,
and a central source of electrons. For the DM profile, we include both
spherical and ellipsoidal DM morphologies and a range of radial profiles from
steep cusps to kiloparsec-sized cores, motivated in part by hydrodynamical
simulations. Our derived upper limits on the dark matter annihilation flux
place strong constraints on DM properties. In the case of the pure $b$-quark
annihilation channel, our limits on the annihilation cross section are more
stringent than those from the Milky Way dwarfs up to DM masses of approximately
TeV and rule out the thermal relic cross section up to approximately 300 GeV.
Better understanding of the DM profile, as well as the Fermi-LAT data at its
highest energies, would further improve the sensitivity to DM properties.
