  We compare the surface brightness profile and morphology of the Galactic
Centre Excess (GCE) identified in wide-angle $\gamma$-ray maps from the
Fermi-Large Area Telescope to dark matter annihilation predictions derived from
high-resolution $\Lambda$CDM magnetohydrodynamic simulations of galaxy
formation. These simulations produce isolated, disc-dominated galaxies with
structure, stellar populations, gas content, and stellar and halo masses
comparable to those of the Milky Way. For a specific choice of annihilation
cross-section, they agree well with the Fermi-LAT data over the full observed
angular range, $1^{\circ}$ to $15^{\circ}$, whereas their dark-matter only
counterparts, lacking any compression of the inner halo by the gravitational
effects of the baryons, fail to predict emission as centrally concentrated as
observed. These results provide additional support to the hypothesis that the
GCE is produced by annihilating dark matter. If, however, it is produced by a
different mechanism, they imply a strong upper limit on annihilation rates
which can be translated into upper limits on the expected $\gamma$-ray flux not
only from the inner Galaxy but also from any substructure, with or without
stars, in the Galactic halo.
