  We consider a closed Riemannian manifold $M$ of negative curvature and
dimension at least 3 with marked length spectrum sufficiently close
(multiplicatively) to that of a locally symmetric space $N$. Using the methods
of Hamenst\"adt, we show the volumes of $M$ and $N$ are approximately equal. We
then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with
derivative bounds close to 1 and depending on the ratio of the two marked
length spectrum functions. Thus, we refine the results of Hamenst\"adt and
Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked
length spectra are equal. We also prove a similar result for compact negatively
curved surfaces using the methods of Otal together with a version of the Gromov
compactness theorem due to Greene--Wu and Pugh.
