Measurements are reported of the central exclusive production of J/ψ and ψ(2S) mesons in pp collisions at a centre-of-mass energy of 13 TeV. Backgrounds are significantly reduced compared to previous measurements made at lower energies through the use of new forward shower counters. The products of the cross-sections and the branching fractions for the decays to dimuons, where both muons are within the pseudorapidity range 2.0 < η < 4.5, are measured to be
σJ/ψ→μ+μ−=435±18±11±17 pbσψ(2S)→μ+μ−=11.1±1.1±0.3±0.4 pb.
The first uncertainties are statistical, the second are systematic, and the third are due to the luminosity determination. The cross-sections are also measured differentially for meson rapidities between 2.0 and 4.5. Good agreement is observed with theoretical predictions. Photoproduction cross-sections are derived and compared to previous experiments, and a deviation from a pure power-law extrapolation of lower energy data is observed.