This article deals with the approximation of the boundary control of the linear one-dimensional wave equation. It is known that the high frequency spurious oscillations that the classical methods of finite difference and finite element introduce lead to nonuniform controllability properties (see [J. A. Infante and E. Zuazua, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 407-438]. A space-discrete scheme with an added numerical vanishing viscous term is introduced and analyzed. The extra numerical damping filters out the high numerical frequencies and ensures the convergence of the sequence of discrete controls to a control of the continuous conservative wave equation when the mesh size tends to zero.