A search has been performed, using the full 20.3 fb−1 data sample of 8 TeV proton--proton collisions collected in 2012 with the ATLAS detector at the LHC, for photons originating from a displaced vertex due to the decay of a neutral long-lived particle into a photon and an invisible particle. The analysis investigates the diphoton plus missing transverse momentum final state, and is therefore most sensitive to pair-production of long-lived particles. The analysis technique exploits the capabilities of the ATLAS electromagnetic calorimeter to make precise measurements of the flight direction, as well as the time of flight, of photons. No excess is observed over the Standard Model predictions for background. Exclusion limits are set within the context of Gauge Mediated Supersymmetry Breaking models, with the lightest neutralino being the next-to-lightest supersymmetric particle and decaying into a photon and gravitino with a lifetime in the range from 250 ps to about 100 ns.