  In this paper we prove that an abelian group contains $(2^{2m+1}(2^{m-1}+1),
2^m(2^m+1), 2^m)$-difference sets with $m\geqslant 3$ if and only if it
contains an elementary abelian 2-group of order $2^{2m}$. Our proof shows that
the method of constructing such difference sets is essentially unique.
