  Gamma-ray data from the Fermi-Large Area Telescope reveal an unexplained,
apparently diffuse, signal from the Galactic bulge that peaks near 2 GeV with
an approximately spherical intensity profile $\propto r^{-2.4}$ that extends to
angular radial scales of at least 10 degrees, possibly to 20 degrees. The
origin of this "Galactic Centre Excess" (GCE) has been debated with proposed
sources prominently including self-annihilating dark matter and a hitherto
undetected population of millisecond pulsars (MSPs). However, the conventional
channel for the generation of MSPs has been found to predict too many low mass
X-ray binary (LMXB) systems and, because of the expected large natal kicks, may
not accommodate the close spatial correspondence between the GCE signal and
stars in the bulge. Here we report a binary population synthesis forward model
that demonstrates that an MSP population arising from the accretion induced
collapse (AIC) of O-Ne white dwarfs in Galactic bulge binaries can naturally
reproduce the morphology, spectral shape, and intensity of the GCE signal while
also obeying LMXB constraints. Synchrotron emission from MSP-launched cosmic
ray electrons and positrons may simultaneously explain the mysterious,
microwave "haze" from the inner Galaxy
