  The double longitudinal-spin asymmetry, $A_{LL}$, of the $J/\psi$ production
in polarized proton-proton collisions is presented in this paper at QCD
next-to-leading order. It is found that the obtained values of $A_{LL}$ are in
general consistent with the PHENIX measurements. Various sets of the
long-distance matrix elements (LDMEs) are employed in our calculation to study
the possible theoretical uncertainties. It is found that, for $p_t<5\gev$, all
these LDMEs lead to almost the same results, which are within the tolerance of
the experimental data uncertainties.
