Cosmic-ray interactions with the atmosphere produce a flux of neutrinos in all directions with energies extending above the TeV scale(1). The Earth is not a fully transparent medium for neutrinos with energies above a few TeV, as the neutrinonucleon cross-section is large enough to make the absorption probability non-negligible(2). Since absorption depends on energy and distance travelled, studying the distribution of the TeV atmospheric neutrinos passing through the Earth offers an opportunity to infer its density profiles(3-7). This has never been done, however, due to the lack of relevant data. Here we perform a neutrino-based tomography of the Earth using actual data-one-year of through-going muon atmospheric neutrino data collected by the IceCube telescope(8). Using only weak interactions, in a way that is completely independent of gravitational measurements, we are able to determine the mass of the Earth and its core, its moment of inertia, and to establish that the core is denser than the mantle. Our results demonstrate the feasibility of this approach to study the Earth's internal structure, which is complementary to traditional geophysics methods. Neutrino tomography could become more competitive as soon as more statistics is available, provided that the sources of systematic uncertainties are fully under control.