  We report on a search for double beta decay of $^{130}$Te to the first
$0^{+}$ excited state of $^{130}$Xe using a 9.8 kg$\cdot$yr exposure of
$^{130}$Te collected with the CUORE-0 experiment. In this work we exploit
different topologies of coincident events to search for both the neutrinoless
and two-neutrino double-decay modes. We find no evidence for either mode and
place lower bounds on the half-lives: $\tau^{0\nu}_{0^+}>7.9\cdot 10^{23}$ yr
and $\tau^{2\nu}_{0^+}>2.4\cdot 10^{23}$ yr. Combining our results with those
obtained by the CUORICINO experiment, we achieve the most stringent constraints
available for these processes: $\tau^{0\nu}_{0^+}>1.4\cdot 10^{24}$ yr and
$\tau^{2\nu}_{0^+}>2.5\cdot 10^{23}$ yr.
