We study e+e-→J/ψcc̄ by measuring the invariant mass distribution recoiling against fully reconstructed J/ψ decays, using 124fb-1 of data collected at a center-of-mass energy of 10.6 GeV with the BABAR detector. We observe signals for ηc(1S), χc0, and ηc(2S) in the recoil mass distribution, thus confirming previous measurements. We measure σ(e+e-→J/ψ+cc̄)B(cc̄→>2charged) to be 17.6±2.8(stat)-2.1+1.5(syst)fb, 10.3±2.5(stat)-1.8+1.4(syst)fb, and 16.4±3.7(stat)-3.0+2.4(syst)fb with cc̄=ηc(1S), χc0, and ηc(2S), respectively. © 2005 The American Physical Society.