We report the first dark matter search results from XENON1T, a ∼2000−kg-target-mass dual-phase (liquid-gas) xenon time projection chamber in operation at the Laboratori Nazionali del Gran Sasso in Italy and the first ton-scale detector of this kind. The blinded search used 34.2 live days of data acquired between November 2016 and January 2017. Inside the (1042±12)−kg fiducial mass and in the [5,40] keVnr energy range of interest for weakly interacting massive particle (WIMP) dark matter searches, the electronic recoil background was (1.93±0.25)×10−4 events/(kg×day×keVee), the lowest ever achieved in such a dark matter detector. A profile likelihood analysis shows that the data are consistent with the background-only hypothesis. We derive the most stringent exclusion limits on the spin-independent WIMP-nucleon interaction cross section for WIMP masses above 10 GeV/c2, with a minimum of 7.7×10−47 cm2 for 35−GeV/c2 WIMPs at 90% C.L.