  An invariant Gibbs' state for the nonlinear Schrodinger equation on the
circle was constructed by Bourgain, and McKean, out of the basic Hamiltonian
using a trigonometric cut-off. The cubic nonlinear Schrodinger equation is a
completely integrable system having an infinite number of additional integrals
of motion. In this paper we construct the second invariant Gibbs' state from
one of these additional integrals for the cubic NLS on the circle. This
additional Gibbs' state is singular with respect to the Gibbs' state previously
constructed from the basic Hamiltonian. Our approach employs the Ablowitz-Ladik
system, a completely integrable discretization of the cubic Schrodinger
equation.
