This paper considers the boundary rigidity problem for a compact convex Riemannian manifold
(
M
,
g
)
(M,g)
with boundary
∂
M
\partial M
whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics
g
′
g’
on
M
M
there is a
C
3
,
α
C^{3,\alpha }
-neighborhood of
g
g
such that
g
g
is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance — as measured in
M
M
). More precisely, given any metric
g
′
g’
in this neighborhood with the same boundary distance function there is diffeomorphism
φ
\varphi
which is the identity on
∂
M
\partial M
such that
g
′
=
φ
∗
g
g’=\varphi ^{*}g
. There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.