  We evaluate the production cross section for direct J/Psi and Upsilon
integrated in P_T for various collision energies in the QCD-based
Colour-Singlet Model (CSM). We consider the LO contribution from gluon fusion
whose P_T-integrated cross section shows a very good agreement with the
Tevatron and LHC data, both for J/Psi and Upsilon. The rapidity distribution of
this yield is evaluated in the central region relevant for the ATLAS and CMS
detectors, as well as in the more forward region relevant for the ALICE and
LHCb detectors. The results obtained here are compatible with those of other
approaches within the range of the theoretical uncertainties which are
admittedly very large. This suggests that the "mere" measurements of the yield
at the LHC will not help disentangle between the different possible quarkonium
production mechanisms. Yet, the comparison with the first LHC results by ALICE,
ATLAS, CMS and LHCb confirms that the CSM correctly accounts for the
P_T-integrated yield at sqrt(s)=7 TeV.
