  In this paper, we study the amount of information contained in the Steklov
spectrum of some compact manifolds with connected boundary equipped with a
warped product metric. Examples of such manifolds can be thought of as deformed
balls in R^d. We first prove that the Steklov spectrum determines uniquely the
warping function of the metric. We show in fact that the approximate knowledge
(in a given precise sense) of the Steklov spectrum is enough to determine
uniquely the warping function in a neighbourhood of the boundary. Second, we
provide stability estimates of log-type on the warping function from the
Steklov spectrum. The key element of these stability results relies on a
formula that, roughly speaking, connects the inverse data (the Steklov
spectrum) to the Laplace transform of the difference of the two warping
factors.
