We report the results of a search for supersymmetry (SUSY) with gauge-mediated breaking in the missing transverse energy distribution of inclusive diphoton events using 263 pb-1 of data collected by the D0 experiment at the Fermilab Tevatron Collider in 2002–2004. No excess is observed above the background expected from standard model processes, and lower limits on the masses of the lightest neutralino and chargino of about 108 and 195 GeV, respectively, are set at the 95% confidence level. These are the most stringent limits to date for models with gauge-mediated SUSY breaking with a short-lived neutralino as the next-to-lightest SUSY particle.