  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}.
