  Let C be a smooth projective complex curve of genus $g\geq2$. We investigate
the Brill-Noether locus consisting of stable bundles of rank 2 and canonical
determinant having at least $k$ independent sections. Using the Hecke
correpondence we construct a fundamental class, which determines the
non-emptiness of this locus at least when $C$ is a Petri curve. We prove that
in many expected cases the Brill-Noether locus is non-empty. For some values of
$k$ the result is best possible.
