  Using tools of quantum information theory we show that the ground state of
the Dicke model exhibits an infinite sequence of instabilities
(quantum-phase-like transitions). These transitions are characterized by abrupt
changes of the bi-partite entanglement between atoms at critical values
$\kappa_j$ of the atom-field coupling parameter $\kappa$ and are accompanied by
discontinuities of the first derivative of the energy of the ground state. We
show that in a weak-coupling limit ($\kappa_1\leq \kappa \leq \kappa_2$) the
Coffman-Kundu-Wootters (CKW) inequalities are saturated which proves that for
these values of the coupling no intrinsic multipartite entanglement (neither
among the atoms nor between the atoms and the field) is generated by the
atom-field interaction. We analyze also the atom-field entanglement and we show
that in the strong-coupling limit the field is entangled with the atoms so that
the von Neumann entropy of the atomic sample (that serves as a measure of the
atom-field entanglement) takes the value $S_A={1/2}\ln (N+1)$. The entangling
interaction with atoms leads to a highly sub-Poissonian photon statistics of
the field mode.
