  Measurements are reported of the central exclusive production of \jpsi and
\psitwos 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 $2.0<\eta<4.5$, are measured to
be $$
  \begin{array}{rcl} \sigma_{J/\psi\rightarrow\mu^+\mu^-}&=&435 \pm 18 \pm 17
\pm 16 {\rm \ pb},\\ \sigma_{\psi(2S)\rightarrow\mu^+\mu^-}&=&11.1 \pm 1.1 \pm
0.3 \pm 0.4 {\rm \ pb}.\\ \end{array} $$ 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.
