  We analytically calculate the cross sections of double charmonium production
in $e^+ e^- \to J/\psi(\psi(2S))\chi_{cJ}$ (J=0,1,2) at next-to-leading order
(NLO) in $\alpha_s$ in nonrelativistic QCD, and confirm factorization of these
processes. In contrast to $\chi_{c0}$ production, for which the NLO correction
is large and positive, the NLO corrections for $\chi_{c1,2}$ production can be
negative, resulting in decreased $K$ factors of 0.91 and 0.78 for J=1 and 2
respectively when $\mu = 2 m_{c}$. Consequently, the NLO QCD corrections
markedly enlarge the difference between cross sections of $\chi_{c0}$ and
$\chi_{c1,2}$. This may explain why $e^+ e^- \to J/\psi(\psi(2S))\chi_{c0}$ but
not $e^+ e^- \to J/\psi(\psi(2S))\chi_{c1,2}$ is observed experimentally.
Moreover, for $J/\psi(\psi(2S))\chi_{c1,2}$, the NLO QCD corrections
substantially reduce the $\mu$ dependence and lead to predictions with small
theoretical uncertainties.
