  We consider bordered Riemann surfaces which are biholomorphic to compact
Riemann surfaces of genus g with n regions biholomorphic to the disc removed.
We define a refined Teichmueller space of such Riemann surfaces and demonstrate
that in the case that 2g+2-n>0, this refined Teichmueller space is a Hilbert
manifold. The inclusion map from the refined Teichmueller space into the usual
Teichmueller space (which is a Banach manifold) is holomorphic.
  We also show that the rigged moduli space of Riemann surfaces with
non-overlapping holomorphic maps, appearing in conformal field theory, is a
complex Hilbert manifold. This result requires an analytic reformulation of the
moduli space, by enlarging the set of non-overlapping mappings to a class of
maps intermediate between analytically extendible maps and quasiconformally
extendible maps. Finally we show that the rigged moduli space is the quotient
of the refined Teichmueller space by a properly discontinuous group of
biholomorphisms.
