  We consider the 1D nonlinear Schr\"odinger equation with bilinear control. In
the case of Neumann boundary conditions, local exact controllability of this
equation near the ground state has been proved by Beauchard and Laurent in
arXiv:1001.3288. In this paper, we study the case of Dirichlet boundary
conditions. To establish the controllability of the linearised equation, we use
a bilinear control acting through four directions: three Fourier modes and one
generic direction. The Fourier modes are appropriately chosen so that they
satisfy a saturation property. These modes allow to control approximately the
linearised Schr\"odinger equation. We show that the reachable set for the
linearised equation is closed. This is achieved by representing the resolving
operator as a sum of two linear continuous mappings: one is surjective (here
the control in generic direction is used) and the other is compact. A mapping
with dense and closed image is surjective, so the linearised Schr\"odinger
equation is exactly controllable. Then local exact controllability of the
nonlinear equation is derived using the inverse mapping theorem.
