  We investigate the ${\cal O}(\alpha_s)$ corrections to $e^+e^-\to
J/\psi(\psi')+\chi_{cJ}$ ($J=0,1,2$) in the NRQCD factorization approach. These
next-to-leading order (NLO) corrections are calculated at the level of helicity
amplitude. We have made a detailed analysis for both polarized and unpolarized
cross sections, and compared our predictions with the measurements at the $B$
factories. We also derive the asymptotic expressions for each of the NLO
helicity amplitudes, and confirm the earlier speculation that at NLO in
$\alpha_s$, the double logarithm of type $\ln^2 (s/m_c^2)$ appearing in the
NRQCD short-distance coefficient is always associated with the
helicity-suppressed channels.
