  The moduli space M(r,d) of stable, rank r, degree d vector bundles on a
smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A,
where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The
birational isomorphism is compatible with fixing determinants in M(r,d) and
M(h,0) and we obtain as a corollary that the moduli space of bundles of rank r
and fixed determinant of degree d is rational, when r and d are coprime. A key
ingredient in the proof is the use of a naturally defined Brauer class for the
function field of M(r,d).
