We determine and study all Lie bialgebra central extensions of $\R^{2n}$ by $\R$ admitting the Heisenberg algebra $\H$ as the underlying Lie algebra structure. The present work answers the question about the realizability of Lie bialgebra structures on $\H$ as central extensions of $\R^{2n}$ endowed with the adapted Lie bialgebra structure, by $\R$.