If the supermassive black hole Sgr A* at the center of the Milky Way grew adiabatically from an initial seed embedded in a Navarro-Frenk-White dark matter (DM) halo, then the DM profile near the hole has steepened into a spike. We calculate the dramatic enhancement to the gamma-ray flux from the Galactic center (GC) from such a spike if the 1-3 GeV excess observed in Fermi data is due to DM annihilations. We find that for the parameter values favored in recent fits, the point-source-like flux from the spike is 35 times greater than the flux from the inner 1° of the halo, far exceeding all Fermi point source detections near the GC. We consider the dependence of the spike signal on astrophysical and particle parameters and conclude that if the GC excess is due to DM, then a canonical adiabatic spike is disfavored by the data. We discuss alternative Galactic histories that predict different spike signals, including (i) the nonadiabatic growth of the black hole, possibly associated with halo and/or black hole mergers, (ii) gravitational interaction of DM with baryons in the dense core, such as heating by stars, or (iii) DM self-interactions. We emphasize that the spike signal is sensitive to a different combination of particle parameters than the halo signal and that the inclusion of a spike component to any DM signal in future analyses would provide novel information about both the history of the GC and the particle physics of DM annihilations.