This article reports a measurement of the production cross section of prompt isolated photon pairs in proton-antiproton collisions at s√=1.96  TeV using the CDF II detector at the Fermilab Tevatron collider. The data correspond to an integrated luminosity of 5.36  fb−1. The cross section is presented as a function of kinematic variables sensitive to the reaction mechanisms. The results are compared with three perturbative QCD calculations: (1) a leading-order parton shower Monte Carlo, (2) a fixed next-to-leading-order calculation and (3) a next-to-leading-order/next-to-next-to-leading-log resummed calculation. The comparisons show that, within their known limitations, all calculations predict the main features of the data, but no calculation adequately describes all aspects of the data.