We have measured the differential production cross section d2σdΩdp for pions, kaons, protons, and antiprotons produced in 12.4-12.5-GeVc proton-proton collisions. We systematically studied the dependence of these cross sections on the center-of-mass longitudinal and transverse momentum of π's, K's, and p̄'s, and on pc.m. and θc.m. for protons, over much of the available phase space. The extracted proton beam of the Argonne zero-gradient synchrotron (ZGS) impinged on a liquid-hydrogen target. The scattered particles were detected by a spectrometer using magnets and scintillation and Cherenkov counters to identify and momentum-analyze these particles. A steering magnet allowed detection of particles produced at different laboratory angles without moving any of the apparatus. The incident proton intensity was determined by monitor counters calibrated using foil spallation techniques. The pion production cross section was well represented by a sum of two Gaussians in p2:d2σdΩdp=Ae-15p2+Be-3p2. The slope for large-momentum-transfer kaon production was also about 3 (GeVc)-2. The pion production cross section as a function of pl c.m. had a maximum at pl c.m.=0, possibly supporting a one-center model. Incorrect earlier measurements supporting a two-center model are discussed and corrected. The inelastic scattering cross section for protons also shows a less prominent forward peak and an e-3p2 dependence. This proton cross section also seems independent of inelasticity over a large range.