J/psi production in p+p collisions at 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 < p(T)< 9 GeV/c. The size of the present data set allows a detailed measurement of both the p(T) and the rapidity distributions and is sufficient to constrain production models. The total cross section times the branching ratio is B-ll sigma(J/psi)(pp)=178 +/- 3(stat)+/- 53(sys)+/- 18(norm) nb.