We search for the rare decay of the D0 meson to two photons, D0→γγ, and present a measurement of the branching fraction for a D0 meson decaying to two neutral pions, B(D0→π0π0). The data sample analyzed corresponds to an integrated luminosity of 470.5 fb-1 collected by the BABAR detector at the PEP-II asymmetric-energy e+e- collider at SLAC. We place an upper limit on the branching fraction, B(D0→γγ)<2.2×10-6, at 90% confidence level. This limit improves on the existing limit by an order of magnitude. We also find B(D0→π0π0)=(8.4±0.1±0.4±0.3)×10-4.