  Previously we gave a proof of the Feigin--Fuchs character formula for the
irreducible unitary discrete series of the Virasoro algebra with 0<c<1. The
proof showed directly that the mutliplicity space arising in the coset
construction of Goddard, Kent and Olive was irreducible, using the elementary
part of the unitarity criterion of Friedan, Qiu and Shenker, giving
restrictions on h for c=1-6/m(m+1) with m>2. In this paper we consider the same
problem in the limiting case of the coset construction c=1. Using primary
fields, we directly establish that the Virasoro algebra acts irreducibly on the
multiplicity spaces of irreducible representations of SU(2) in the two level
one irreducible representations of the corresponding affine Kac--Moody algebra.
This gives a direct proof that the only singular vectors in these
representations are those given by Goldstone's formulas, which also play an
important part in the proof. For this proof, the theory is developed from
scratch in a self-contained semi-expository way. Using the Jantzen filtration
and the Kac determinant formula, we give an additional independent proof for
the case c=1 which generalises to the case 0<c<1, where it provides an
alternative approach to that of Astashkevich.
