We present the results of a search for anomalous production of diphoton events with large missing transverse energy using the Collider Detector at Fermilab. In 202 \invpb of \ppbar collisions at $\sqrt{s}=1.96$ TeV we observe no candidate events, with an expected standard model background of $0.27\pm0.07({\rm stat})\pm0.10({\rm syst})$ events. The results exclude a lightest chargino of mass less than 167 GeV/$c^2$, and lightest neutralino of mass less than 93 GeV/$c^2$ at 95% C.L. in a gauge--mediated supersymmetry-- breaking model with a light gravitino.