Evidence for an extraterrestrial flux of high-energy neutrinos has now been found in multiple searches with the IceCube detector. The first solid evidence was provided by a search for neutrino events with deposited energies $\gtrsim30$ TeV and interaction vertices inside the instrumented volume. Recent analyses suggest that the extraterrestrial flux extends to lower energies and is also visible with throughgoing, $��_��$-induced tracks from the Northern hemisphere. Here, we combine the results from six different IceCube searches for astrophysical neutrinos in a maximum-likelihood analysis. The combined event sample features high-statistics samples of shower-like and track-like events. The data are fit in up to three observables: energy, zenith angle and event topology. Assuming the astrophysical neutrino flux to be isotropic and to consist of equal flavors at Earth, the all-flavor spectrum with neutrino energies between 25 TeV and 2.8 PeV is well described by an unbroken power law with best-fit spectral index $-2.50\pm0.09$ and a flux at 100 TeV of $\left(6.7_{-1.2}^{+1.1}\right)\cdot10^{-18}\,\mathrm{GeV}^{-1}\mathrm{s}^{-1}\mathrm{sr}^{-1}\mathrm{cm}^{-2}$. Under the same assumptions, an unbroken power law with index $-2$ is disfavored with a significance of 3.8 $��$ ($p=0.0066\%$) with respect to the best fit. This significance is reduced to 2.1 $��$ ($p=1.7\%$) if instead we compare the best fit to a spectrum with index $-2$ that has an exponential cut-off at high energies. Allowing the electron neutrino flux to deviate from the other two flavors, we find a $��_e$ fraction of $0.18\pm0.11$ at Earth. The sole production of electron neutrinos, which would be characteristic of neutron-decay dominated sources, is rejected with a significance of 3.6 $��$ ($p=0.014\%$).

16 pages, 10 figures; accepted for publication in The Astrophysical Journal; updated one reference