  A finite group $G$ is called a Schur group if any $S$-ring over $G$ is
associated in a natural way with a subgroup of $Sym(G)$ that contains all right
translations. We prove that the groups $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$,
where $n\geq 1$, are Schur. Modulo previously obtained results, it follows that
every noncyclic Schur $p$-group, where $p$ is an odd prime, is isomorphic to
$\mathbb{Z}_3\times \mathbb{Z}_3 \times \mathbb{Z}_3$ or $\mathbb{Z}_3\times
\mathbb{Z}_{3^n}$, $n\geq 1$ .
