We present a new measurement of the e(+)e(-)-> J/psi c(c)over bar cross section where the c(c)over bar pair can fragment either into charmed hadrons or a charmonium state. In the former case the J/psi and a charmed hadron are reconstructed, while the latter process is measured using the recoil mass technique, which allows the identification of two-body final states without reconstruction of one of the charmonia. The measured e(+)e(-)-> J/psi c(c)over bar cross section is (0.74 +/- 0.08(-0.08)(+0.09)) pb, and the e(+)e(-)-> J/psi Xnon-c(c)over bar cross section is (0.43 +/- 0.09 +/- 0.09) pb. We note that the measured cross sections are obtained from a data sample with the multiplicity of charged tracks in the event larger than 4; corrections for the effect of this requirement are not performed as this cannot be done in a model-independent way. The analysis is based on a data sample with an integrated luminosity of 673 fb(-1) recorded near the Upsilon(4S) resonance with the Belle detector at the KEKB e(+)e(-) asymmetric-energy collider.