Search for the Decay D0 -> gamma gamma and Measurement of the Branching Fraction for D0 -> pi0 pi0

We search for the rare decay of the $D^{0}$ meson to two photons, $D^0 \rightarrow \gamma \gamma$, and present a measurement of the branching fraction for a $D^0$ meson decaying to two neutral pions, ${B}(D^0 \rightarrow \pi^0 \pi^0)$. The data sample analyzed corresponds to an integrated luminosity of 470.5 {\sl B {-0.4em} {\small\sl A} {-0.37em} \sl B {-0.4em} {\small\sl A {-0.02em}R}} detector at the PEP-II asymmetric-energy $e^{+} e^{-}$ collider at SLAC. We place an upper limit on the branching fraction for the decay of a $D^0$ meson to two photons, ${B}(D^0 \rightarrow \gamma \gamma) < 2.2 \times 10^{-6}$, at 90% confidence level, and find ${B}(D^0 \rightarrow \pi^0 \pi^0) = (8.4 \pm 0.1 \pm 0.4 \pm 0.3) \times 10^{-4}$.