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: $^{130}$Te neutrinoless double-beta decay. Examining a total TeO$_2$ exposure of 86.3 kg$cdot$yr, characterized by an effective energy resolution of (7.7 $pm$ 0.5) keV FWHM and a background in the region of interest of (0.014 $pm$ 0.002) counts/(keV$cdot$kg$cdot$yr), we find no evidence for neutrinoless double-beta decay. The median statistical sensitivity of this search is $7.0times10^{24}$ yr. Including systematic uncertainties, we place a lower limit on the decay half-life of $T^{0 u}_{1/2}$($^{130}$Te) > $1.3times 10^{25}$ yr (90% C.L.). Combining this result with those of two earlier experiments, Cuoricino and CUORE-0, we find $T^{0 u}_{1/2}$($^{130}$Te) > $1.5times 10^{25}$ 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 $m_{betabeta}<(110 - 520)$ meV, where the range reflects the nuclear matrix element estimates employed.
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