From the introduction: The category \(\text{TF}_R\) of finite rank torsion-free modules over a discrete valuation ring \(R\) with prime \(p\) is known to be complex. For instance, if \(S\) is a finite antichain, then there are uncountably many different embeddings of the category \(\text{rep}(S,R)\) of finite rank free \(R\)-representations of \(S\) into \(\text{TF}_R\). The complexity of \(\text{TF}_R\) is demonstrated by the fact that if \(|S|\geq 3\), then \(\text{rep}(S,R)\) has wild mod \(p\) representation type. On the other hand, purely indecomposable modules (pure finite rank \(R\)-submodules of the completion \(R^*\) of \(R)\) are easily characterized up to isomorphism. Consequently, co-purely indecomposable modules (quasi-isomorphic duals of purely indecomposable modules) are characterized up to quasi-isomorphism. This article is devoted to a classification of co-purely indecomposable modules up to isomorphism.