  We compute the rational cohomology groups of the smooth Brill-Noether
varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension
exactly $r$ on a general curve $C$. As an application, we determine the whole
intersection cohomology of the singular Brill-Noether loci $W^r_d(C)$,
parametrizing complete linear series on $C$ of degree $d$ and dimension at
least $r$.
