The differential cross-section as a function of rapidity has been measured for the exclusive production of J/psi and psi(2S) mesons in proton-proton collisions at root s = 7 TeV, using data collected by the LHCb experiment, corresponding to an integrated luminosity of 930 pb(-1). The cross-sections times branching fractions to two muons having pseudorapidities between 2.0 and 4.5 are measured to be sigma (pp -> J/psi -> mu+mu-)(2.0 < eta(mu +/-) < 4.5) = 291 +/- 7 +/- 19 pb, sigma (pp ->psi(2S)->mu+mu-)(2.0 < eta(mu +/-) < 4.5) = 6.5 +/- 0.9 +/- 0.4 pb, where the first uncertainty is statistical and the second is systematic. The measurements agree with next-to-leading order QCD predictions as well as with models that include saturation effects.