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 ν_{τ} events were found with visible energies ranging from roughly 20 TeV to 1 PeV and a median expected parent ν_{τ} 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-ν_{τ} astrophysical neutrinos. Thus, we rule out the absence of astrophysical ν_{τ} at the 5σ level. The measured astrophysical ν_{τ} flux is consistent with expectations based on previously published IceCube astrophysical neutrino flux measurements and neutrino oscillations.