Production cross-sections of prompt charm mesons are measured using data from pp collisions at the LHC at a centre-of-mass energy of 5 TeV. The data sample corresponds to an integrated luminosity of 8.60 ± 0.33 pb−1 collected by the LHCb experiment. The production cross-sections of D0, D+, Ds +, and D∗+ mesons are measured in bins of charm meson transverse momentum, pT, and rapidity, y. They cover the rapidity range 2.0 < y < 4.5 and transverse momentum ranges 0 < pT< 10 GeV/c for D0 and D+ and 1 < pT< 10 GeV/c for Ds + and D∗+ mesons. The inclusive cross-sections for the four mesons, including charge-conjugate states, within the range of 1 < pT< 8 GeV/c are determined to be(Formula Presented.)where the uncertainties are statistical and systematic, respectively.[Figure not available: see fulltext.].