 Solution

Every model of $T$ becomes an $L_0$-structure by forgetting the symbols outside $L_0$, so $T_0$ has an infinite model. If $\phi$ is an $L_0$-sentence, completeness of $T$ gives either $T\models \phi$ or $T\models \neg \phi$. Accordingly either $\phi \in T_0$ or $\neg \phi \in T_0$. The two cannot both occur because $T$ is consistent. Therefore $T_0$ is a complete $L_0$-theory with infinite models.