CERN-LHC. Measurement of the production of J/PSI mesons in proton-proton collisions at a centre-of-mass energy of 7 TeV. The data sample has a total integrated luminosity of 314 +- 34 nb-1. Total cross sections (within the kinematical limits of PT from 6.5 to 30 GeV and |y|<2.4)) and differential cross sections are determined for J/PSIs produced either promptly or non-promptly. For the non-prompt J/PSIs, produced through B-hadron decay, various polarization scenarios are considerd, (0), (-1,CS), (+1,CS), (-1,HX), (+1,HX), where (0) is unpolarized, CS(HX) is the Collins-Soper (Helicity) frame and -1(+1) refer to full longitudinal (transverse) polarization. The cross sections are for the J/PSI to MU+MU- decay mode and uncorrected for the branching ratio.