A search for dark-matter particles in events with large missing transverse momentum and a Higgs boson candidate decaying into two photons is reported. The search uses 139 fb⁻¹ of proton-proton collision data collected at s = 13 TeV with the ATLAS detector at the CERN LHC between 2015 and 2018. No significant excess of events over the Standard Model predictions is observed. The results are interpreted by extracting limits on three simplified models that include either vector or pseudoscalar mediators and predict a final state with a pair of dark-matter candidates and a Higgs boson decaying into two photons. [Figure not available: see fulltext.]