This Letter reports a measurement of the cross section for producing pairs of central prompt isolated photons in proton-antiproton collisions at a total energy s√=1.96  TeV using data corresponding to 9.5  fb−1 integrated luminosity collected with the CDF II detector at the Fermilab Tevatron. The measured differential cross section is compared to three calculations derived from the theory of strong interactions. These include a prediction based on a leading order matrix element calculation merged with a parton shower model, a next-to-leading order calculation, and a next-to-next-to-leading order calculation. The first and last calculations reproduce most aspects of the data, thus showing the importance of higher-order contributions for understanding the theory of strong interaction and improving measurements of the Higgs boson and searches for new phenomena in diphoton final states.