Using data samples of 102 x 10(6) gamma(1S) and 158 x 10(6) gamma(2S) events collected with the Belle detector, a first experimental search has been made for double-charmonium production in the exclusive decays gamma(1S, 2S) --> J/psi(psi') + X, where X = eta(c).chi(cJ) (J = 0, 1, 2), eta(c)(2S), X(3940), and X(4160). No significant signal is observed in the spectra of the mass recoiling against the reconstructed J/psi or psi' except for the evidence of chi(c1) production with a significance of 4.6 sigma for gamma(1S) -> J/psi + chi(c1). The measured branching fraction B(gamma(1S) -> J/psi + chi(c1)) is [3.90 +/- 1.21 (stat) +/- 0.23(syst) x 10(-6). The 90% C.L. upper limits on the branching fractions of the other modes having a significance of less than 3 sigma are determined. These results are consistent with theoretical calculations using the nonrelativistic QCD factorization approach.