We present measurements from events with two isolated prompt photons in pbar-p collisions at $\sqrt{s} = 1.8$ TeV. The data were accumulated to an integrated luminosity of 4.3 1/pb. Dominant backgrounds to prompt photons are single neutral mesons decaying into multiple photons, which are dissociated from jets. To subtract the backgrounds, the photon identification by the lateral shower profile measured with the strip chamber embedded in the central electromagnetic calorimeter is applied to the data. We obtain 49 +- 15(stat) +20-14(syst) events in the final diphoton data set. The diphoton events include 1.8 +- 0.6 jets with transverse energy of 4.5 +- 1.9 GeV on average, but no large missing transverse energy. Thus the two-photon system have transverse momentum of 5.1 +- 1.1 GeV/c on average. The event topology of diphoton production is comparable with that of W boson production. The cross section for photons with transverse momentum pT in the range of 10 < pT < 29 [GeV/c] is 95 +- 28(stat)+38-27(syst) pb. The differential cross section, measured as a function of pT of each photon, is about 3 times what a next-to-leading-logarithm calculation predicts.