This paper investigates torsion-free abelian groupsAwhich areQE-flat, i.e. for whichQAis flat as anQE(A)-module. It is shown that a torsion-freeAhas this property iff Tor1(M, A)is torsion for all rightE(A)-modulesM. Furthermore, a torsion-free group of rank 4 is constructed which isQE-flat but not quasi-isomorphic to anE-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which areQE-flat.