A search for diphoton events with large missing transverse momentum has been performed using 1.07 fb^-1 of proton-proton collision data at sqrt(s) = 7 TeV recorded with the ATLAS detector. No excess of events was observed above the Standard Model prediction and 95% Confidence Level (CL) upper limits are set on the production cross section for new physics: sigma <(22-129) fb in the context of a generalised model of gauge-mediated supersymmetry breaking (GGM) with a bino-like lightest neutralino, sigma <(27-91) fb in the context of a minimal model of gauge-mediated supersymmetry breaking (SPS8) and (15-27) fb in the context of a specific model with one universal extra dimension (UED). A 95% CL lower limit of 805 GeV, for bino masses above 50 GeV, is set on the GGM gluino mass. Lower limits of 145 TeV and 1.23 TeV are set on the SPS8 breaking scale Lambda and on the UED compactification scale 1/R, respectively. These limits provide the most stringent tests of these models to date.