  By the description due to Mukai and Iliev, a smooth prime Fano threefold X of
genus 9 is associated to a surface P(V), ruled over a smooth plane quartic
Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X
with odd determinant. For each c_2 \geq 7, we prove that a component of their
moduli space M_X(2,1,c_2) is birational to a Brill-Noether locus of bundles on
Gamma having enough sections when twisted by V. Moreover we prove that
M_X(2,1,7) is isomorphic to the blowing-up of the Picard variety Pic^2(Gamma)
along the curve parametrizing lines contained in X.
