Neutrinos interact only very weakly, so they are extremely penetrating. The theoretical neutrino-nucleon interaction cross-section, however, increases with increasing neutrino energy, and neutrinos with energies above 40 teraelectronvolts (TeV) are expected to be absorbed as they pass through the Earth. Experimentally, the cross-section has been determined only at the relatively low energies (below 0.4 TeV) that are available at neutrino beams from accelerators. Here we report a measurement of neutrino absorption by the Earth using a sample of 10,784 energetic upward-going neutrino-induced muons. The flux of high-energy neutrinos transiting long paths through the Earth is attenuated compared to a reference sample that follows shorter trajectories. Using a fit to the two-dimensional distribution of muon energy and zenith angle, we determine the neutrino-nucleon interaction cross-section for neutrino energies 6.3-980 TeV, more than an order of magnitude higher than previous measurements. The measured cross-section is about 1.3 times the prediction of the standard model, consistent with the expectations for charged- and neutral-current interactions. We do not observe a large increase in the cross-section with neutrino energy, in contrast with the predictions of some theoretical models, including those invoking more compact spatial dimensions or the production of leptoquarks. This cross-section measurement can be used to set limits on the existence of some hypothesized beyond-standard-model particles, including leptoquarks.