  We present the first complete study of $\Upsilon$ and prompt $J/\psi$
production from single-parton scattering, including the complete
$\mathcal{O}(\alpha_S^6)$ color-singlet contribution, the
$\mathcal{O}(\alpha_S^2\alpha^2)$ electroweak contribution, the complete
nonrelativistic S-wave and P-wave color-octet contribution as well as the
feeddown contribution. Our study was motivated by the recent evidence reported
by D0 Collaboration of prompt $J/\psi$ and $\Upsilon$ simultaneous production
at the Tevatron. With our complete evaluation, we are able to refine the
determination of the double parton scattering contribution made by D0
Collaboration. We find that the effective cross section characterizing the
importance of double-parton scatterings is $\sigma_{\rm eff}\le 8.2$ mb at
$68\%$ confidence level from the D0 measurement.
