We search for the rare decay of the D^0 meson to two photons, D^0→γγ, and present a measurement of the branching fraction for a D^0 meson decaying to two neutral pions, B(D^0→π^0π^0). 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(D^0→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(D^0→π^0π^0)=(8.4±0.1±0.4±0.3)×10^(-4)
