We consider a damped wave equation on a open subset of R n or a smooth Riemannian manifold with boundary, with Ventcel boundary conditions, with a linear damping, acting either in the interior or at the boundary. This equation is a model for a vibrating structure with a layer with higher rigidity of thickness δ > 0. By means of a proper Carleman estimate for second-order elliptic operators near the boundary, we derive a resolvent estimate for the wave semigroup generator along the imaginary axis, which in turn yields the logarithmic decay rate of the energy. This stabilization result is obtained uniformly in δ.