  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 ${B}(D^0 \rightarrow \pi^0 \pi^0) =
(8.4 \pm 0.1 \pm 0.4 \pm 0.3) \times 10^{-4}$.
