  The character of the exceptional series of representations of SU(1,1) is
determined by using Bargmann's realization of the representation in the Hilbert
space $H_\sigma$ of functions defined on the unit circle. The construction of
the integral kernel of the group ring turns out to be especially involved
because of the non-local metric appearing in the scalar product with respect to
which the representations are unitary. Since the non-local metric disappears in
the `momentum space' $i.e.$ in the space of the Fourier coefficients the
integral kernel is constructed in the momentum space, which is transformed back
to yield the integral kernel of the group ring in $H_\sigma$. The rest of the
procedure is parallel to that for the principal series treated in a previous
paper. The main advantage of this method is that the entire analysis can be
carried out within the canonical framework of Bargmann.
