We search for inclusive high ${E}_{T}$ diphoton events with large missing transverse energy in $p\overline{p}$ collisions at $\sqrt{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8\phantom{\rule{0ex}{0ex}}\mathrm{TeV}$. Such events are expected from pair production of charginos and neutralinos within the framework of the minimal supersymmetric standard model with a light gravitino. No excess of events is observed. In that model, and assuming gaugino mass unification at the GUT scale, we obtain a 95% C.L. exclusion region in the supersymmetry parameter space and lower mass bounds of $150\mathrm{GeV}{/c}^{2}$ for the lightest chargino and $77\mathrm{GeV}{/c}^{2}$ for the lightest neutralino.