Using γ-ray data from the Fermi Large Area Telescope, various groups have identified a clear excess emission in the Inner Galaxy, at energies around a few GeV. This excess resembles remarkably well a signal from dark-matter annihilation. One of the most compelling astrophysical interpretations is that the excess is caused by the combined effect of a previously undetected population of dim γ-ray sources. Because of their spectral similarity, the best candidates are millisecond pulsars. Here, we search for this hypothetical source population, using a novel approach based on wavelet decomposition of the γ-ray sky and the statistics of Gaussian random fields. Using almost seven years of Fermi-LAT data, we detect a clustering of photons as predicted for the hypothetical population of millisecond pulsar, with a statistical significance of 10.0σ. For plausible values of the luminosity function, this population explains 100% of the observed excess emission. We argue that other extragalactic or Galactic sources, a mismodeling of Galactic diffuse emission, or the thick-disk population of pulsars are unlikely to account for this observation.