  We present the results of a search for supersymmetry with gauge-mediated
breaking and $\NONE\to\gamma\Gravitino$ in the $\gamma\gamma$+missing
transverse energy final state. In 2.6$\pm$0.2 \invfb of $p{\bar p}$ collisions
at $\sqrt{s}$$=$1.96 TeV recorded by the CDF II detector we observe no
candidate events, consistent with a standard model background expectation of
1.4$\pm$0.4 events. We set limits on the cross section at the 95% C.L. and
place the world's best limit of 149 \gevc on the \none mass at
$\tau_{\tilde{\chi}_1^0}$$\ll$1 ns. We also exclude regions in the \none\
mass-lifetime plane for $\tau_{\tilde{\chi}_1^0}$$\lesssim$2 ns.
