  The associated production of prompt $J/\psi$ and $\mathit{\mathit{\Upsilon}}$
mesons in $pp$ collisions at a centre-of-mass energy of
$\sqrt{s}=13\,\mathrm{TeV}$ is studied using LHCb data, corresponding to an
integrated luminosity of $4\,\mathrm{fb}^{-1}$. The measurement is performed
for $J/\psi$ ($\mathit{\Upsilon}$) mesons with a transverse momentum
$p_{\mathrm{T}}<10\,(30)\,\mathrm{GeV}/c$ in the rapidity range $2.0<y<4.5$. In
this kinematic range, the cross-section of the associated production of prompt
$J/\psi$ and $\mathit{\Upsilon}(1S)$ mesons is measured to be $133 \pm 22 \pm 7
\pm 3 \, \mathrm{pb}$, with a significance of $7.9\,\sigma$, and that of prompt
$J/\psi$ and $\mathit{\Upsilon}(2S)$ mesons to be $76\pm 21 \pm 4 \pm 7 \,
\mathrm{pb}$, with a significance of $4.9\,\sigma$. 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 $J/\psi$ and $\mathit{\Upsilon}(1S)$ in proton-proton collisions.
Differential cross-sections are measured as functions of variables that are
sensitive to kinematic correlations between the $J/\psi$ and
$\mathit{\Upsilon}(1S)$ mesons. The effective cross-sections of the associated
production of prompt $J/\psi$ and $\mathit{\Upsilon}$ mesons are obtained and
found to be compatible with measurements using other particle productions.
