The production of $\mathit{\Xi}_{cc}^{++}$ baryons in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=13$ TeV is measured in the transverse-momentum range $4<p_\mathrm{T}<15~\mathrm{GeV}$/$c$ and the rapidity range $2.0<y<4.5$. The data used in this measurement correspond to an integrated luminosity of 1.7 $\mathrm{fb}^{-1}$, recorded by the LHCb experiment during 2016. The ratio of the $\mathit{\Xi}_{cc}^{++}$ production cross-section times the branching fraction of the $\mathit{\Xi_{cc}^{++} \to \Lambda_{c}^{+} K^- \pi^+ \pi^+}$ decay relative to the prompt $\mathit{\Lambda_c^{+}}$ production cross-section is found to be $(2.22\pm 0.27 \pm 0.29)\times 10^{-4}$, assuming the central value of the measured $\mathit{\Xi_{cc}^{++}}$ lifetime, where the first uncertainty is statistical and the second systematic.