We search for the rare decay of the D-0 meson to two photons, D-0 -> gamma gamma, and present a measurement of the branching fraction for a D-0 meson decaying to two neutral pions, B(D-0 -> pi(0)pi(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 -> gamma gamma) < 2.2 x 10(-6), at 90% confidence level. This limit improves on the existing limit by an order of magnitude. We also find B(D-0 -> pi(0)pi(0)) = (8.4 +/- 0.1 +/- 0.4 +/- 0.3 +/-) x 10(-4).
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