  Let $(X, \omega)$ be a weakly pseudoconvex K\"ahler manifold, $Y \subset X$ a
closed submanifold defined by some holomorphic section of a vector bundle over
$X,$ and $L$ a Hermitian line bundle satisfying certain positivity conditions.
We prove that for any integer $k\geq 0,$ any section of the jet sheaf $L\otimes
{\cal O}_X/{\cal I}_Y^{k+1},$ which satisfies a certain $L^2$ condition, can be
extended into a global holomorphic section of $L$ over $X$ whose $L^2$ growth
on an arbitrary compact subset of $X$ is under control. In particular, if $Y$
is merely a point, this gives the existence of a global holomorphic function
with an $L^2$ norm under control and with prescribed values for all its
derivatives up to order $k$ at a point. This result generalizes the $L^2$
extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of
sections of a line bundle. A technical difficulty is to achieve uniformity in
the constant appearing in the final estimate. In this respect, we make use of
the exponential map and of a Rauch-type comparison theorem for complete
Riemannian manifolds.
