We present the results of a search for supersymmetry with gauge-mediated breaking and chi(1)(0) --> gammaG in the gammagamma + missing transverse energy final state. In 2.6+/-0.2 fb(-1) of pp collisions at square root(s) = 1.96 TeV recorded by the CDF II detector we observe no candidate events, consistent with a standard model background expectation of 1.4+/-0.4 events. We set limits on the cross section at the 95% C.L. and place the world's best limit of 149 GeV/c2 on the chi(1)(0) mass at tau(chi(1)(0)) << 1 ns. We also exclude regions in the chi(1)(0) mass-lifetime plane for tau(chi(1)(0)) approximately < 2 ns.