  Within the framework of nonrelativisitic QCD factorization formalism, we
present the next-to-leading-order relativistic corrections to $\Upsilon$
exclusive decay into $\eta_c$ plus $J/\psi$. The double charmonia can be
produced through several immediate channels, i.e. $\Upsilon\to g^*g^*g^*\to
\eta_c+J/\psi$, $\Upsilon\to g^*g^*\gamma^*\to \eta_c+J/\psi$, and $\Upsilon\to
\gamma^*\to \eta_c+J/\psi$. The amplitudes of these three channels are obtained
accurate up to ${\cal O}(\alpha_s^3 v^2)$, ${\cal O}(\alpha\alpha_s^2 v^2)$,
and ${\cal O}(\alpha\alpha_s^2 v^2)$ respectively, where $v$ indicates the
typical heavy quark velocity in bottomonium and/or charmonium rest frame. The
decay rates are also presented. We find the next-to-leading-order relativistic
corrections to the short distance coefficients as well as the decay rates are
both significant and negative, especially for the corrections from bottomonium.
More seriously, the decay rates are even brought into negative by including the
relativistic corrections, which indicates the poor convergence for the velocity
expansion in this kind of processes. Detailed analysis is given in the paper.
