The associated production of prompt J/ψ and Υ mesons in pp collisions at a centre-of-mass energy of ss ​ = 13 TeV is studied using LHCb data, corresponding to an integrated luminosity of 4 fb−1−1. The measurement is performed for J/ψ (Υ) mesons with a transverse momentum pTT​< 10 (30) GeV/c in the rapidity range 2.0 < y < 4.5. In this kinematic range, the cross-section of the associated production of prompt J/ψ and Υ(1S) mesons is measured to be 133 ± 22 ± 7 ± 3 pb, with a significance of 7.9 σ, and that of prompt J/ψ and Υ(2S) mesons to be 76 ± 21 ± 4 ± 7 pb, with a significance of 4.9 σ. The first uncertainty is statistical, the second systematic, and the third due to uncertainties on the used branching fractions. This is the first observation of the associated production of J/ψ and Υ(1S) in proton-proton collisions. Differential cross-sections are measured as functions of variables that are sensitive to kinematic correlations between the J/ψ and Υ(1S) mesons. The effective cross-sections of the associated production of prompt J/ψ and Υ mesons are obtained and found to be compatible with measurements using other particle productions.[graphic not available: see fulltext]