  In this article we prove a differentiable rigidity result. Let $(Y, g)$ and
$(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$)
and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that
the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of
$(X,g_0)$. We show that there exists a number $\varepsilon:=\varepsilon (n,
d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by
$-n(n-1)$ and its volume satisfies $\vol_g (Y)\leqslant (1+\varepsilon)
\vol_{g_0} (X)$ then the manifolds are diffeomorphic. The proof relies on
Cheeger-Colding's theory of limits of Riemannian manifolds under lower Ricci
curvature bound.
