  Polarizations of the prompt $J/\psi$'s produced in the $e^+e^-$ annihilation
at CLEO energy ($\sqrt{s} = 10.58$ GeV) are considered in the framework of
NRQCD factorization formalism. We find that the $J/\psi$ polarization has
strong dependence on the production mechanism. At CLEO energy, the most
dominant $J/\psi$ production mechanism in the most phase space is the
color-singlet mechanism, $e^+e^- \to J/\psi+gg$, for which $J/\psi$'s are
highly longitudinally polarized. On the other hand, the dominant $J/\psi$
production mechanism at the upper end point of $J/\psi$ energy distribution is
the color-octet process, $e^+e^- \to (c\bar{c})^{(8)} +g$, for which $J/\psi$'s
are almost unpolarized. Thus, the measurement of the polarization of the end
point $J/\psi$'s can give another test of color-octet mechanism, and constraint
on the NRQCD matrix elements.
