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. Including systematic uncertainties, we place a lower limit on the decay half-life of T0ν1/2 ( 130Te ) > 1.3 × 1025 yr (90% C.L.); the median statistical sensitivity of this search is 7.0 × 1024 yr. Combining this result with those of two earlier experiments, Cuoricino and CUORE-0, we find T0ν1/2 ( 130Te ) > 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 mββ < ( 110 − 520 ) meV , where the range reflects the nuclear matrix element estimates employed.