  We consider the one dimensional Schr\"odinger equation with a bilinear
control and prove the rapid stabilization of the linearized equation around the
ground state. The feedback law ensuring the rapid stabilization is obtained
using a transformation mapping the solution to the linearized equation on the
solution to an exponentially stable target linear equation. A suitable
condition is imposed on the transformation in order to cancel the non-local
terms arising in the kernel system. This conditions also insures the uniqueness
of the transformation. The continuity and invertibility of the transformation
follows from exact controllability of the linearized system.
