  We consider rigidity properties of compact symmetric spaces $X$ with metric
$g_0$ of rank one. Suppose $g$ is another Riemannian metric on $X$ with
sectional curvature $\kappa$ bounded by $0 \leq \kappa \leq 1$. If $g$ equals
$g_0$ outside a convex proper subset of $X$, then $g$ is isometric with $g_0$.
We also exhibit examples of surfaces showing that the nonnegativity of the
curvature is needed. Our main result complements earlier results on other
symmetric spaces by Gromov and Schroeder-Ziller.
