  Using data samples of $102\times10^6$ $\Upsilon(1S)$ and $158\times10^6$
$\Upsilon(2S)$ events collected with the Belle detector, a first experimental
search has been made for double-charmonium production in the exclusive decays
$\Upsilon(1S,2S)\rightarrow J/\psi(\psi')+X$, 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 $\Upsilon(1S)\rightarrow J/\psi+\chi_{c1}$. The
measured branching fraction $\BR(\Upsilon(1S)\rightarrow J/\psi+\chi_{c1})$ is
$(3.90\pm1.21(\rm stat.)\pm0.23 (\rm syst.))\times10^{-6}$. The $90\%$
confidence level 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.
