Neutrinos interact only very weakly, so they are extremely penetrating. However, the theoretical neutrino-nucleon interaction cross section rises with energy such that, at energies above 40 TeV, neutrinos are expected to be absorbed as they pass through the Earth. Experimentally, the cross section has been measured only at the relatively low energies (below 400 GeV) available at neutrino beams from accelerators cite{Agashe:2014kda, Formaggio:2013kya}. Here we report the first measurement of neutrino absorption in the Earth, using a sample of 10,784 energetic upward-going neutrino-induced muons observed with the IceCube Neutrino Observatory. The flux of high-energy neutrinos transiting long paths through the Earth is attenuated compared to a reference sample that follows shorter trajectories through the Earth. Using a fit to the two-dimensional distribution of muon energy and zenith angle, we determine the cross section for neutrino energies between 6.3 TeV and 980 TeV, more than an order of magnitude higher in energy than previous measurements. The measured cross section is $1.30^{+0.21}_{-0.19}$ (stat.) $^{+0.39}_{-0.43}$ (syst.) times the prediction of the Standard Model cite{CooperSarkar:2011pa}, consistent with the expectation for charged and neutral current interactions. We do not observe a dramatic increase in the cross section, expected in some speculative models, including those invoking new compact dimensions cite{AlvarezMuniz:2002ga} or the production of leptoquarks cite{Romero:2009vu}.
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