The IceCube Neutrino Observatory has observed a diffuse flux of TeV-PeVastrophysical neutrinos at 5.7 sigma significance from an all-flavor search. The direct detection of tau neutrinos in this flux has yet to occur. Tau neutrinos become distinguishable from other flavors in IceCube at energies above a few hundred TeV, when the cascade from the tau neutrino charged current interaction becomes resolvable from the cascade from the tau lepton decay. This paper presents results from the first dedicated search for tau neutrinos with energies between 214 TeV and 72 PeV in the full IceCube detector. The analysis searches for IceCube optical sensors that observe two separate pulses in a single event-one from the tau neutrino interaction and a second from the tau decay. No candidate events were observed in three years of IceCube data. For the first time, a differential upper limit on astrophysical tau neutrinos is derived around the PeV energy region, which is nearly 3 orders of magnitude lower in energy than previous limits from dedicated tau neutrino searches.