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 $5.1\pm 0.3{\rm~keV}$ 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 $2.9\times 10^{24}~{\rm yr}$ 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, $T^{0\nu}_{1/2}>$~$ 2.7\times 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 $T^{0\nu}_{1/2} > 4.0\times 10^{24}~\mathrm{yr}$ 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, $m_{\beta\beta}< 270$ -- $760~\mathrm{meV}$.
