We report on a search method for astrophysical tau neutrinos with 9.7 years of IceCube data acquired between 2011 and 2020. We use convolutional neural networks (CNNs) to detect the "double pulse" signature created by the closely-separated particle showers deposited at the tau neutrino interaction vertex and subsequent tau lepton decay vertex. In contrast to previous IceCube tau neutrino searches that looked for double pulse signatures on one or two digital optical modules (DOMs), the CNNs use 180 DOMs on three neighboring IceCube strings. The predicted peak sensitivity of the analysis is at tau neutrino energies of roughly 200 TeV, lower than previous searches and thereby conferring increased sensitivity to astrophysical tau neutrinos whose flux falls as E_nu^{-2}. We present the seven tau neutrino candidates identified by our search and discuss the expected backgrounds to these events