A measurement of prompt photon-pair production in proton-proton collisions at √s = 13 TeV is presented. The data were recorded by the ATLAS detector at the LHC with an integrated luminosity of 139 fb⁻¹. Events with two photons in the well-instrumented region of the detector are selected, and the photons are required to be isolated and have E_{T, γ_1(2)} > 40 (30) GeV for the leading (sub-leading) photon. The differential cross section as functions of several observables for the diphoton system are measured in the fiducial volume and compared to theoretical predictions from state-of-the-art Monte Carlo and fixed-order calculations. The measured integrated cross section is compatible with the predictions from next-to-next-to-leading order and multileg-merged calculations. Predictions from lower-orders in perturbative QCD fail to agree with data within their estimated theoretical uncertainties. Agreement between measured differential cross sections and the most accurate predictions is generally good where expected, showing a clear necessity to take higher-order perturbative QCD corrections into account.