A bstract A measurement of prompt photon-pair production in proton-proton collisions at $$ \sqrt{s} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt></mml:math> = 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. The photons are required to be isolated and have a transverse momentum of $$ {p}_{\mathrm{T}{,}_{\gamma 1(2)}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi>γ</mml:mi><mml:mn>1</mml:mn><mml:mfenced><mml:mn>2</mml:mn></mml:mfenced></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math> &gt; 40 (30) GeV for the leading (sub-leading) photon. The differential cross sections as functions of several observables for the diphoton system are measured and compared with theoretical predictions from state-of-the-art Monte Carlo and fixed-order calculations. The QCD predictions from next-to-next-to-leading-order calculations and multi-leg merged calculations are able to describe the measured integrated and differential cross sections within uncertainties, whereas lower-order calculations show significant deviations, demonstrating that higher-order perturbative QCD corrections are crucial for this process. The resummed predictions with parton showers additionally provide an excellent description of the low transverse-momentum regime of the diphoton system.