Let $\mathcal T_n$ denote the set of all labelled spanning trees of $K_n$. A family $\mathcal F \subset \mathcal T_n$ is $t$-intersecting if for all $A, B \in \mathcal F$ the trees $A$ and $B$ share at least $t$ edges. In this paper, we determine for $n>n_0$ the size of the largest $t$-intersecting family $\mathcal F\subset \mathcal T_n$ for all meaningful values of $t$ ($t\le n-1$). This result is a rare instance when a complete $t$-intersection theorem for a given type of structures is known.