  Let $C$ be a nonsingular projective curve of genus $g\ge2$ defined over the
complex numbers, and let $M_{\xi}$ denote the moduli space of stable bundles of
rank $n$ and determinant $\xi$ on $C$, where $\xi$ is a line bundle of degree
$d$ on $C$ and $n$ and $d$ are coprime. It is shown that the universal bundle
$\cu_{\xi}$ on $C\times M_{\xi}$ is stable with respect to any polarisation on
$C\times M_{\xi}$. It is shown further that the connected component of the
moduli space of $\cu_{\xi}$ containing $\cu_{\xi}$ is isomorphic to the
Jacobian of $C$.
