We study tensor completions \({G\otimes }_{{\mathcal{N}}_{2,R}}R\) of finitely generated torsion-free 2-nilpotent groups G in the class \({\mathcal{N}}_{2,R}\) of all 2-nilpotent R-groups over a binomial domain R. We show that \({G\otimes }_{{\mathcal{N}}_{2,R}}R\) is isomorphic to the group ( \({G\otimes }_{\mathcal{H}}R\) ) × D, where \({G\otimes }_{\mathcal{H}}R\) is the classical Hall R-completion of G, D is an Abelian R-group, and the direct product is a product of abstract groups (not R-groups!). In particular, this answers an old question of Remeslennikov about the algebraic structure of free 2-nilpotent R-groups in the quasivariety \({\mathcal{N}}_{2,R}\) (they are precisely the tensor R-completions of free 2-nilpotent groups).
