Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97].