The IceCube Neutrino Observatory was designed primarily to search for high-energy (TeV-PeV) neutrinos produced in distant astrophysical objects. A search for ≳100  TeV neutrinos interacting inside the instrumented volume has recently provided evidence for an isotropic flux of such neutrinos. At lower energies, IceCube collects large numbers of neutrinos from the weak decays of mesons in cosmic-ray air showers. Here we present the results of a search for neutrino interactions inside IceCube’s instrumented volume between 1 TeV and 1 PeV in 641 days of data taken from 2010–2012, lowering the energy threshold for neutrinos from the southern sky below 10 TeV for the first time, far below the threshold of the previous high-energy analysis. Astrophysical neutrinos remain the dominant component in the southern sky down to a deposited energy of 10 TeV. From these data we derive new constraints on the diffuse astrophysical neutrino spectrum, Φ_ν=2.06_{-0.3}^{+0.4}×10-18(E_ν/10^5  GeV)^{-2.46±0.12} GeV^-1 cm^−2 sr^−1 s^-1 for 25  TeV<E_ν<1.4  PeV, as well as the strongest upper limit yet on the flux of neutrinos from charmed-meson decay in the atmosphere, 1.52 times the benchmark theoretical prediction used in previous IceCube results at 90% confidence.