Making use of 36 pb^-1 of proton-proton collision data at sqrt{s} = 7 TeV, the ATLAS Collaboration has performed a search for diphoton events with large missing transverse energy. Observing no excess of events above the Standard Model prediction, a 95% Confidence Level (CL) upper limit is set on the cross section for new physics of sigma < 0.38 - 0.65 pb in the context of a generalised model of gauge mediated supersymmetry breaking (GGM) with a bino-like lightest neutralino, and of sigma < 0.18 - 0.23 pb in the context of a specific model with one universal extra dimension (UED). A 95 % CL lower limit of 560 GeV, for bino masses above 50 GeV, is set on the GGM gluino mass, while a lower limit of 1/R > 961 GeV is set on the UED compactification radius R. These limits provide the most stringent tests of these models to date.