  The group Diff(S^1) of the orientation preserving diffeomorphisms of the
circle S^1 plays an important role in conformal field theory. We consider a
subgroup B_0 of Diff(S^1) whose elements stabilize "the point of infinity".
This subgroup is of interest for the actual physical theory living on the
punctured circle, or the real line. We investigate the unique central extension
K of the Lie algebra of that group. We determine the first and second
cohomologies, its ideal structure and the automorphism group. We define a
generalization of Verma modules and determine when these representations are
irreducible. Its endomorphism semigroup is investigated and some unitary
representations of the group which do not extend to Diff(S^1) are constructed.
