The possibility of measuring cross sections for the production of J/ψ mesons in fixed-target experiments with the proton and ion beams of the Large Hadron Collider (LHC) at CERN (Switzerland) is considered. At the present time, measurements of charmonium production in proton-proton collisions at an energy of 7 TeV have begun at LHC. Previously, the production of J/ψ and ψ′ mesons was studied in the NA38, NA50, and NA60 fixed-target experiments with beams of the CERN synchrotron (SPS) and in the PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC, Brookhaven National Laboratory, USA). A normal nuclear absorption of J/ψ mesons in proton-nucleus collisions and an enhanced, anomalous, suppression of the production of charmonium states in central collisions of relativistic nuclei were observed. At the present time, there are no theoretical models that could describe the entire body of experimental data. Measurements over a broad interval of proton and ion energies are required. Measurements of charmonium production using LHC beams with fixed targets in the energy range between the SPS and RHIC energies-a beam of 7-TeV protons (√s = 114.6 GeV) and a beam of 2.75-TeV/nucleon lead ions (√s = 71.8 GeV)-will provide an additional possibility for studying the charmonium-production mechanism. Estimates of the geometric acceptance, luminosity, and counting rate for the production of J/ψ mesons are presented.