© 2014 American Physical Society. Using data samples of 102×106 (1S) and 158×106 (2S) events collected with the Belle detector, a first experimental search has been made for double-charmonium production in the exclusive decays (1S,2S)→J/ψ(ψ′)+X, where X=ηc, χcJ(J=0,1,2), ηc(2S), X(3940), and X(4160). No significant signal is observed in the spectra of the mass recoiling against the reconstructed J/ψ or ψ′ except for the evidence of χc1 production with a significance of 4.6σ for (1S)→J/ψ+χc1. The measured branching fraction B((1S)→J/ψ+χc1) is [3.90±1.21(stat)±0.23(syst)]×10-6. The 90% C.L. upper limits on the branching fractions of the other modes having a significance of less than 3σ are determined. These results are consistent with theoretical calculations using the nonrelativistic QCD factorization approach.