J/psi production in p+p collisions at square root s=200 GeV has been measured by the PHENIX experiment at the BNL Relativistic Heavy Ion Collider over a rapidity range of -2.2<y<2.2 and a transverse momentum range of 0<pT<9 GeV/c. The size of the present data set allows a detailed measurement of both the pT and the rapidity distributions and is sufficient to constrain production models. The total cross section times the branching ratio is Bll sigma pp J/psi=178+/-3stat+/-53sys+/-18norm nb.