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 (similar to 90%) by electron and tau flavors. The flux, observed in the sensitive energy range from 16 TeV to 2.6 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 +/- 0.07 and a flux normalization for each neutrino flavor of phi(astro) = 1.66(-0.27)(+0.25) at E-0 = 100 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 <= 2.28 at >= 3 sigma significance level. Because of high neutrino energy resolution and low atmospheric neutrino backgrounds, this analysis provides the most detailed characterization of the neutrino flux at energies below similar to 100 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 >= 0.06). The sizable and smooth flux measured below similar to 100 TeV remains a puzzle. In order to not violate the isotropic diffuse gamma-ray background as measured by the Fermi Large Area Telescope, it suggests the existence of astrophysical neutrino sources characterized by dense environments which are opaque to gamma rays.