In this paper we are concerned with the output regulation problem for a system (plant) described by a boundary controlled anti-stable one-dimensional Schrödinger equation by the backstepping approach. The output to be controlled is not required to be measurable and its output operator is assumed to be admissible for a certain C0-semigroup that is related to the original system whereas the measurement is located at the boundary. First we show that the open-loop system is well-posed. We construct a state feedback control law to solve the output regulation problem, using the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on these observers, we design an output feedback controller that achieves output regulation.