  In nonrelativistic QCD (NRQCD), we study $e^+ e^- \to J/\psi(\psi(2S))
+\chi_{c0}$ at $B$ factories, where the P-wave state $\chi_{c0}$ is associated
with an S-wave state $J/\psi$ or $\psi(2S)$. In contrast to the failure of
factorization in most cases involving P-wave states, e.g. in B decays, we find
that factorization holds in this process at next to leading order (NLO) in
$\alpha_s$ and leading order (LO) in $v$, where the associated S-wave state
plays a crucial rule in canceling the infrared (IR) divergences. We also give
some general analyses for factorization in various double charmonium
production. The NLO corrections in $e^+ e^- \to J/\psi(\psi(2S)) +\chi_{c0}$ at
$\sqrt{s}=10.6$ GeV are found to substantially enhance the cross sections by a
factor of about 2.8; hence crucially reduce the large discrepancy between
theory and experiment. With $m_c=1.5{\rm GeV}$ and $\mu=2m_c$, the NLO cross
sections are estimated to be $17.9(11.3)$ fb for $e^+ e^- \to J/\psi(\psi(2S))+
\chi_{c0}$, which reach the lower bounds of experiment.
