  We consider the low-energy effective action for the Coulomb phase of an $N =
2$ supersymmetric gauge theory with a rank one gauge group. The $N = 2$
superspace formalism is naturally invariant under an $SL(2, {\bf Z})$ group of
duality transformations, regardless of the form of the action. The leading and
next to leading terms in the long distance expansion of the action are given by
the holomorphic prepotential and a real analytic function respectively. The
latter is shown to be modular invariant with respect to $SL(2, {\bf Z})$.
