We consider the one-dimensional nonlinear Schr\"odinger equation with
Dirichlet boundary conditions in the fully resonant case (absence of the
zero-mass term). We investigate conservation of small amplitude
periodic-solutions for a large set measure of frequencies. In particular we
show that there are infinitely many periodic solutions which continue the
linear ones involving an arbitrary number of resonant modes, provided the
corresponding frequencies are large enough and close enough to each other (wave
packets with large wave number)
