  We report on the inclusive J/$\psi$ production cross section measured at the
CERN Large Hadron Collider in proton-proton collisions at a centre-of-mass
energy $\sqrt{s}~=~13$ TeV. The J/$\psi$ mesons are reconstructed in the $\rm
e^{+} e^{-}$ decay channel and the measurements are performed at midrapidity
($|y|<0.9$) in the transverse-momentum interval $0< p_{\rm T} <40$ GeV/$c$,
using a minimum-bias data sample corresponding to an integrated luminosity
$L_{\text{int}} = 32.2~\text{nb}^{-1}$ and an Electromagnetic Calorimeter
triggered data sample with $L_{\text{int}} = 8.3~\mathrm{pb}^{-1}$. The $p_{\rm
T}$-integrated J/$\psi$ production cross section at midrapidity, computed using
the minimum-bias data sample, is $\text{d}\sigma/\text{d}y|_{y=0} =
8.97\pm0.24~(\text{stat})\pm0.48~(\text{syst})\pm0.15~(\text{lumi})~\mu\text{b}$.
An approximate logarithmic dependence with the collision energy is suggested by
these results and available world data, in agreement with model predictions.
The integrated and $p_{\rm T}$-differential measurements are compared with
measurements in pp collisions at lower energies and with several recent
phenomenological calculations based on the non-relativistic QCD and Color
Evaporation models.
