  Using a data sample corresponding to an integrated luminosity of
$2.0\,fb^{-1}$, collected by the LHCb experiment, the production of the
$\eta_c(1S)$ state in proton-proton collisions at a centre-of-mass energy of
$\sqrt{s}=13 \text{ TeV}$ is studied in the rapidity range ${2.0 < y < 4.5}$
and in the transverse momentum range ${6.5 < p_{T} < 14.0\text{ GeV}}$. The
cross-section for prompt production of $\eta_c(1S)$ mesons relative to that of
the $J/\psi$ meson is measured using the ${p\bar{p}}$ decay mode and is found
to be ${\sigma_{\eta_c(1S)}/\sigma_{J/\psi} = 1.69 \pm 0.15 \pm 0.10 \pm
0.18}$. The quoted uncertainties are, in order, statistical, systematic and due
to uncertainties on the branching fractions of the ${J/\psi\to p \bar{p}}$ and
${\eta_c\to p \bar{p}}$ decays. The prompt $\eta_c(1S)$ production
cross-section is determined to be ${\sigma_{\eta_c(1S)} = 1.26 \pm 0.11\pm 0.08
\pm 0.14 \,\mu b}$, where the last uncertainty includes that on the ${J/\psi}$
meson cross-section. The ratio of the branching fractions of $b$-hadron decays
to the $\eta_c(1S)$ and ${J/\psi}$ states is measured to be
${\mathcal{B}_{b\to\eta_c X}/\mathcal{B}_{b\to J/\psi X} = 0.48 \pm 0.03 \pm
0.03 \pm 0.05}$, where the last uncertainty is due to those on the branching
fractions of the ${J/\psi \to p \bar{p}}$ and ${\eta_c\to p \bar{p}}$ decays.
The difference between the ${J/\psi}$ and $\eta_c(1S)$ masses is also
determined to be ${113.0 \pm 0.7 \pm 0.1\text{ MeV}}$, which is the most
precise single measurement of this quantity to date.
