  Let $C$ be a smooth projective complex curve of genus $g \geq 2$. We
investigate the Brill-Noether locus consisting of stable bundles of rank 2 and
determinant $L$ of odd degree $d$ having at least $k$ independent sections.
This locus possesses a virtual fundamental class. We show that in many cases
this class is non-zero, which implies that the Brill-Noether locus is
non-empty. For many values of $d$ and $k$ the result is best possible. We
obtain more precise results for $k\le5$. An appendix contains the proof of a
combinatorial lemma which we need.
