  The inclusive $J/\psi$ production in $e^+ e^- \to J/\psi c\bar c$ at B
factories is one of the most challenging open problems in heavy quarkonium
physics. The observed cross section of this double charm production process is
larger than existing leading order (LO) QCD predictions by a factor of 5. In
the nonrelativistic QCD (NRQCD) factorization formalism, we calculate the
next-to-leading order (NLO) QCD virtual and real corrections to this process,
and find that these corrections can substantially enhance the cross section
with a $K$ factor of about 1.8. We further take into account the feeddown
contributions from higher charmonium states (mainly the $\psi(2S)$ as well as
$\chi_{cJ}$) and the two-photon contributions, and find that the discrepancy
between theory and experiment can be largely removed.
