This Letter reports a measurement of the cross section of prompt isolated photon pair production in pp collisions at a total energy s√=1.96  TeV using data of 5.36  fb−1 integrated luminosity collected with the CDF II detector at the Fermilab Tevatron. The measured cross section, differential in basic kinematic variables, is compared with three perturbative QCD predictions, a leading order parton shower calculation and two next-to-leading order calculations. The next-to-leading order calculations reproduce most aspects of the data. By including photon radiation from quarks before and after hard scattering, the parton shower prediction becomes competitive with the next-to-leading order predictions.