Fraction field of group ring of field over torsion-free abelian group

Question

Let $G$ be a torsion-free abelian group. If $k$ is a field, it is known that $k[G]$ is an integral domain. Let $k(G)=\operatorname{Frac} k[G]$. If $G,H$ are torsion-free abelian groups such that $k(G) \cong k(H) $ for every field $k$, then is it true that $G\cong H $ ?