Let $R$ be a ring. An $R$-module $M$ is called a (weak) duo module provided every (direct summand) submodule of $M$ is fully invariant. It is proved that if $R$ is a commutative domain with field of fractions $K$ then a torsion-free uniform $R$-module is a duo module if and only if every element $k$ in $K$ such that $kM$ is contained in $M$ belongs to $R$. Moreover every non-zero finitely generated torsion-free duo $R$-module is uniform. In addition, if $R$ is a Dedekind domain then a torsion $R$-module is a duo module if and only if it is a weak duo module and this occurs precisely when the $P$-primary component of $M$ is uniform for every maximal ideal $P$ of $R$.