  We prove that the Birkhoff map $\Om$ for KdV constructed on $H^{-1}_0(\T)$
can be interpolated between $H^{-1}_0(\T)$ and $L^2_0(\T)$. In particular, the
symplectic phase space $H^{1/2}_0(\T)$ can be described in terms of Birkhoff
coordinates. As an application, we characterize the regularity of a potential
$q\in H^{-1}(\T)$ in terms of the decay of the gap lengths of the periodic
spectrum of Hill's operator on the interval $[0,2]$.
