The production of J/psi mesons in continuum e(+)e(-) annihilations has been studied with the BABAR detector at energies near the Upsilon(4S) resonance. The mesons are distinguished from J/psi production in B decays through their center-of-mass momentum and energy. We measure the cross section e(+)e(-)-->J/psi X to be 2.52+/-0.21+/-0.21 pb. We set a 90% C.L. upper limit on the branching fraction for direct Upsilon(4S)-->J/psi X decays at 4.7 x 10(-4).