We report the results of a search for neutrinoless double-beta decay in a 9.8~kg$cdot$yr exposure of $^{130}$Te using a bolometric detector array, CUORE-0. The characteristic detector energy resolution and background level in the region of interest are $5.1pm 0.3{rm~keV}$ FWHM and $0.058 pm 0.004,(mathrm{stat.})pm 0.002,(mathrm{syst.})$~counts/(keV$cdot$kg$cdot$yr), respectively. The median 90%~C.L. lower-limit sensitivity of the experiment is $2.9times 10^{24}~{rm yr}$ and surpasses the sensitivity of previous searches. We find no evidence for neutrinoless double-beta decay of $^{130}$Te and place a Bayesian lower bound on the decay half-life, $T^{0 u}_{1/2}>$~$ 2.7times 10^{24}~{rm yr}$ at 90%~C.L. Combining CUORE-0 data with the 19.75~kg$cdot$yr exposure of $^{130}$Te from the Cuoricino experiment we obtain $T^{0 u}_{1/2} > 4.0times 10^{24}~mathrm{yr}$ at 90%~C.L.~(Bayesian), the most stringent limit to date on this half-life. Using a range of nuclear matrix element estimates we interpret this as a limit on the effective Majorana neutrino mass, $m_{betabeta}< 270$ -- $760~mathrm{meV}$.
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