The CUORE experiment, a ton-scale cryogenic bolometer array, recently began operation at the Laboratori Nazionali del Gran Sasso in Italy. The array represents a significant advancement in this technology, and in this work we apply it for the first time to a high-sensitivity search for a lepton-number–violating process: ¹³⁰Te neutrinoless double-beta decay. Examining a total TeO₂ exposure of 86.3 kg⋅yr, characterized by an effective energy resolution of (7.7 ± 0.5) keV FWHM and a background in the region of interest of (0.014 ± 0.002) counts/(keV⋅kg⋅yr), we find no evidence for neutrinoless double-beta decay. The median statistical sensitivity of this search is 7.0 × 10²⁴ yr. Including systematic uncertainties, we place a lower limit on the decay half-life of T_1∕2^0ν > 1.3 × 10²⁵ yr (90% C.L.). Combining this result with those of two earlier experiments, Cuoricino and CUORE-0, we find T_1∕2^0ν > 1.5 × 10²⁵ yr (90% C.L.), which is the most stringent limit to date on this decay. Interpreting this result as a limit on the effective Majorana neutrino mass, we find m_ββ < 140–400 meV, where the range reflects the nuclear matrix element estimates employed.