  In this paper we study the stability of the unique continuation in the case
of the wave equation with variable coefficients independent of time. We prove a
logarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where all
the parameters are calculated explicitly in terms of the $C^1$-norm of the
coefficients and on the other geometric properties of the problem. We use the
Carleman-type estimate proved by Tataru in 1995 and an iteration for locals
stability. We apply the result to the case of a wave equation with data on a
cylinder an we get a stable estimate for any positive time, also after the
first conjugate point for the geodesics of the metric related to the variable
coefficients.
