  We report on the first measurement of the astrophysical neutrino flux using
particle showers (cascades) in IceCube data from 2010 -- 2015. Assuming
standard oscillations, the astrophysical neutrinos in this dedicated cascade
sample are dominated ($\sim 90 \%$) by electron and tau flavors. The flux,
observed in the sensitive energy range from $16\,\mathrm{TeV}$ to
$2.6\,\mathrm{PeV}$, is consistent with a single power-law model as expected
from Fermi-type acceleration of high energy particles at astrophysical sources.
We find the flux spectral index to be $\gamma=2.53\pm0.07$ and a flux
normalization for each neutrino flavor of $\phi_{astro} = 1.66^{+0.25}_{-0.27}$
at $E_{0} = 100\, \mathrm{TeV}$, in agreement with IceCube's complementary muon
neutrino results and with all-neutrino flavor fit results. In the measured
energy range we reject spectral indices $\gamma\leq2.28$ at $\ge3\sigma$
significance level. Due to high neutrino energy resolution and low atmospheric
neutrino backgrounds, this analysis provides the most detailed characterization
of the neutrino flux at energies below $\sim100\,{\rm{TeV}}$ compared to
previous IceCube results. Results from fits assuming more complex neutrino flux
models suggest a flux softening at high energies and a flux hardening at low
energies (p-value $\ge 0.06$). The sizable and smooth flux measured below $\sim
100\,{\rm{TeV}}$ remains a puzzle. In order to not violate the isotropic
diffuse gamma-ray background as measured by the Fermi-LAT, it suggests the
existence of astrophysical neutrino sources characterized by dense environments
which are opaque to gamma-rays.
