The IceCube Neutrino Observatory was designed primarily to search for high-energy (TeV--PeV) neutrinos produced in distant astrophysical objects. A search for $gtrsim 100$~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 IceCubes 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 10 TeV. From these data we derive new constraints on the diffuse astrophysical neutrino spectrum, $Phi_{ u} = 2.06^{+0.4}_{-0.3} times 10^{-18} left({E_{ u}}/{10^5 ,, rm{GeV}} right)^{-2.46 pm 0.12} {rm {GeV^{-1} , cm^{-2} , sr^{-1} , s^{-1}} } $, 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.
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