This article considers the linear 1-d Schrödinger equation in (0,π)perturbed by a vanishing viscosity term depending on a small parameterε > 0. We study the boundary controllability properties of thisperturbed equation and the behavior of its boundary controlsvε as ε goes to zero. Itis shown that, for any time T sufficiently large but independent ofε and for each initial datum inH−1(0,π), there exists a uniformly boundedfamily of controls(vε)ε inL2(0, T) acting on the extremityx = π. Any weak limit of this family is a control forthe Schrödinger equation.