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A search for forbidden and exotic Z boson decays in the diphoton massspectrum is presented for the first time in hadron collisions, based on datacorresponding to 10.0 fb^-1 of integrated luminosity from proton-antiprotoncollisions at sqrt(s) = 1.96 TeV collected by the CDF experiment. No evidenceof signal is observed, and 95% credibility level Bayesian upper limits are seton the branching ratios of decays of the Z boson to a photon and neutral pion(which is detected as a photon), a pair of photons, and a pair of neutralpions. The observed branching ratio limits are 2.01 x 10^-5 for Z to\pi^0\gamma, 1.46 x 10^-5 for Z to \gamma\gamma, and 1.52 x 10^-5 for Z to\pi^0\pi^0. The Z to \pi^0\gamma\ and Z to \gamma\gamma\ limits improve byfactors of 2.6 and 3.6, respectively, the most stringent results from otherexperiments. The Z to \pi^0\pi^0 branching ratio limit is the firstexperimental result on this decay.

First Search for Exotic Z Boson Decays into Photons and Neutral Pions in Hadron Collisions