Exclusive $J/psi$ and $psi(2S)$ vector meson production has been observed in the dimuon channel using the LHCb detector. The cross-section times branching fractions to two muons with pseudorapidities between 2.0 and 4.5 are measured to be $ sigma_{pprightarrow J/psi (rightarrow mu^{+} mu^{-})} (2.0 <eta_{mu^{pm}}< 4.5) = 307 pm 21 pm 36 text{pb}, sigma_{pprightarrow psi(2S) (rightarrow mu^{+} mu^{-})} (2.0 <eta_{mu^{pm}}< 4.5) = 7.8 pm 1.3 pm 1.0 text{pb}, $ where the first uncertainties are statistical and the second are systematic. The measurements are found to be in good agreement with results from previous experiments and theoretical predictions. The $J/psi$ photoproduction cross-section has been measured as a function of the photon-proton centre-of-mass energy. The results are consistent with measurements obtained at HERA and confirm a similar power law behaviour for the photoproduction cross-section.