We report on a measurement of astrophysical tau neutrinos with 9.7 yr of IceCube data. Using convolutional neural networks trained on images derived from simulated events, seven candidate ντ events were found with visible energies ranging from roughly 20 TeV to 1 PeV and a median expected parent ντ energy of about 200 TeV. Considering backgrounds from astrophysical and atmospheric neutrinos, and muons from π±/K± decays in atmospheric air showers, we obtain a total estimated background of about 0.5 events, dominated by non-ντ astrophysical neutrinos. Thus, we rule out the absence of astrophysical ντ at the 5σ level. The measured astrophysical ντ flux is consistent with expectations based on previously published IceCube astrophysical neutrino flux measurements and neutrino oscillations.