The ATLAS experiment at the LHC has measured the production cross section of events with two isolated photons in the final state, in proton-proton collisions at root s = 7 TeV. The full data set collected in 2011, corresponding to an integrated luminosity of 4.9 fb(-1), is used. The amount of background, from hadronic jets and isolated electrons, is estimated with data-driven techniques and subtracted. The total cross section, for two isolated photons with transverse energies above 25 GeV and 22 GeV respectively, in the acceptance of the electromagnetic calorimeter (vertical bar eta vertical bar < 1.37 and 1.52 < vertical bar eta vertical bar 2.37) and with an angular separation Delta R > 0.4, is 44.0(-4.2)(+3.2) pb. The differential cross sections as a function of the di-photon invariant mass, transverse momentum, azimuthal separation, and cosine of the polar angle of the largest transverse energy photon in the Collins-Soper di-photon rest frame are also measured. The results are compared to the prediction of leading-order parton-shower and next-to-leading-order and next-to-next-to-leading-order parton-level generators.