We consider a closed hyperbolic manifold $(N,h)$ of dimension $n\geq 3$ and a manifold $(M,g)$ with a degre one map $f:M \to N$. We will show that if $Ricci_g \geq -(n-1)$ and $Vol (M,g) \leq (1+\epsilon) Vol (N,h)$, then the manifolds $M$ and $N$ are diffeomorphic. The proof relies on Cheeger-Colding theory of limits of Riemannian manifolds under lower Ricci curvature bound.