The cross-section for inelastic proton-proton collisions, with at least one prompt long-lived charged particle of transverse momentum $p_{rm T}>0.2$ GeV/$c$ in the pseudorapidity range $2.0<eta<4.5$, is measured by the LHCb experiment at a centre-of-mass energy of $sqrt{s}=7$ TeV. The cross-section in this kinematic range is determined to be $sigma_{rm inel}^{rm acc} = 55.0 pm 2.4$ mb within the spectrometer acceptance with an experimental uncertainty that is dominated by systematic contributions. Extrapolation to the full phase space, using PYTHIA 6, yields $sigma_{rm inel} = 66.9 pm 2.9 pm 4.4$ mb, where the first uncertainty is experimental and the second is due to the extrapolation.
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