There is evidence for a 130 GeV gamma-ray line at the Galactic Center in the Fermi Large Area Telescope data. Dark matter candidates that explain this feature should also annihilate to Standard Model particles, resulting in a continuous spectrum of photons. To study this continuum, we analyze the Fermi data down to 5 GeV, restricted to the inner 3 degrees of the Galaxy. We place a strong bound on the ratio of continuum photons to monochromatic line photons that is independent of uncertainties in the dark matter density profile. The derived constraints exclude neutralino dark matter as an explanation for the line.