  Inclusive cross sections for $J/\psi$ production in proton-proton collisions
were calculated in the $k_t$-factorization approach for the RHIC energy.
Several mechanisms were considered, including direct color-singlet mechanism,
radiative decays of $\chi_c$ mesons, decays of $\psi'$, open-charm associated
production of $J/\psi$ as well as weak decays of B mesons. Different
unintegrated gluon distributions from the literature were used. We find that
radiative $\chi_c$ decays and direct color-singlet contributions constitute the
dominant mechanism of $J/\psi$ production. These process cannot be consistently
treated within collinear-factorization approach. The results are compared with
recent RHIC data. The new precise data at small transverse momenta impose
stringent constraints on UGDFs. Some UGDFs are inconsistent with the new data.
The Kwieci\'nski UGDFs give the best description of the data. In order to
verify the mechanism suggested here we propose $J/\psi$ -- jet correlation
measurement and an independent measurement of $\chi_c$ meson production in
$\pi^+ \pi^-$ and/or $K^+ K^-$ decay channels. Finally, we address the issue of
\J spin alignment.
