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 D 0 , D + , D s + , and D ∗+ mesons are measured in bins of charm meson transverse momentum, p T , and rapidity, y . They cover the rapidity range 2 . 0 < y < 4 . 5 and transverse momentum ranges 0 < p T < 10 GeV /c for D 0 and D + and 1 < p T < 10 GeV /c for D s + and D ∗+ mesons. The inclusive cross-sections for the four mesons, including charge-conjugate states, within the range of 1 < p T < 8 GeV /c are determined to be σ pp → D 0 X = 1004 ± 3 ± 54 μ b , σ pp → D + X = 402 ± 2 ± 30 μ b , σ pp → D s + X = 170 ± 4 ± 16 μ b , σ pp → D ∗ + X = 421 ± 5 ± 36 μ b , $$ \begin{array}{l}\sigma \left( pp\to {D}^0X\right)=1004\pm 3\pm 54\mu \mathrm{b},\\ {}\sigma \left( pp\to {D}^{+}X\right)=402\pm 2\pm 30\mu \mathrm{b},\\ {}\sigma \left( pp\to {D}_s^{+}X\right)=170\pm 4\pm 16\mu \mathrm{b},\\ {}\sigma \left( pp\to {D}^{\ast +}X\right)=421\pm 5\pm 36\mu \mathrm{b},\end{array} $$ where the uncertainties are statistical and systematic, respectively.