The XENON1T experiment is currently in the commissioning phase at the Laboratori Nazionali del Gran Sasso, Italy. In this article we study the experiment's expected sensitivity to the spin-independent WIMP-nucleon interaction cross section, based on Monte Carlo predictions of the electronic and nuclear recoil backgrounds. The total electronic recoil background in 1 tonne fiducial volume and (1, 12) keV electronic recoil equivalent energy region, before applying any selection to discriminate between electronic and nuclear recoils, is (1.80+/-0.15) . 10(-4) (kg.day.keV)(-1), mainly due to the decay of Rn-222 daughters inside the xenon target. The nuclear recoil background in the corresponding nuclear recoil equivalent energy region (4, 50) keV, is composed of (0.6 +/- 0.1) (t.y)(-1) from radiogenic neutrons, (1.8+/-0.3) . 10(-2) (t.y)(-1) from coherent scattering of neutrinos, and less than 0.01 (t.y)(-1) from muon-induced neutrons. The sensitivity of XENON1T is calculated with the Pro file Likelihood Ratio method, after converting the deposited energy of electronic and nuclear recoils into the scintillation and ionization signals seen in the detector. We take into account the systematic uncertainties on the photon and electron emission model, and on the estimation of the backgrounds, treated as nuisance parameters. The main contribution comes from the relative scintillation efficiency L-eff, which affects both the signal from WIMPs and the nuclear recoil backgrounds. After a 2 y measurement in 1 tonne fiducial volume, the sensitivity reaches a minimum cross section of 1.6 . 10(-47) cm(2) at m(chi) = 50 GeV/c(2).