We report on a search for double beta decay of ¹³⁰Te to the first 0⁺ excited state of ¹³⁰Xe using a 9.8 kg yr exposure of ¹³⁰Te 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 beta decay modes. We find no evidence for either mode and place lower bounds on the half-lives: T^0ν_0⁺1 > 7.9×10²³ yr and T^2ν_0⁺1 > 2.4×10²³ (90% CL). Combining our results with those obtained by the CUORICINO experiment, we achieve the most stringent constraints available for these processes: T^0ν_0⁺1 > 1.4×10²⁴ yr and T^2ν_0⁺1 > 2.5×10²³ yr (90% CL).