  The cross-section for inelastic proton-proton collisions at a centre-of-mass
energy of 13\,TeV is measured with the LHCb detector. The fiducial
cross-section for inelastic interactions producing at least one prompt
long-lived charged particle with momentum $p>2$\,GeV/$c$ in the pseudorapidity
range $2<\eta<5$ is determined to be $\sigma_{\rm acc}= 62.2 \pm 0.2 \pm
2.5$\,mb. The first uncertainty is the intrinsic systematic uncertainty of the
measurement, the second is due to the uncertainty on the integrated luminosity.
The statistical uncertainty is negligible. Extrapolation to full phase space
yields the total inelastic proton-proton cross-section $\sigma_{\rm inel}= 75.4
\pm 3.0 \pm 4.5$\,mb, where the first uncertainty is experimental and the
second due to the extrapolation. An updated value of the inelastic
cross-section at a centre-of-mass energy of 7\,TeV is also reported.
