The flux of high-energy neutrinos passing through the Earth is attenuated due to their interactions with matter. The interaction rate is determined by the neutrino interaction cross section and affects the flux arriving at the IceCube Neutrino Observatory, a cubic-kilometer neutrino detector embedded in the Antarctic ice sheet. We present a measurement of the neutrino cross section between 60 TeV and 10 PeV using the high-energy starting event (HESE) sample from IceCube with 7.5 years of data. The result is binned in neutrino energy and obtained using both Bayesian and frequentist statistics. We find it compatible with predictions from the Standard Model. While the cross section is expected to be flavor independent above 1 TeV, additional constraints on the measurement are included through updated experimental particle identification (PID) classifiers, proxies for the three neutrino flavors. This is the first such measurement to use a ternary PID observable and the first to account for neutrinos from tau decay.