A mechanism for stabilization of the dilaton field within the framework of low-energy string gravitation with loop corrections to the dilaton coupling function was proposed in the first part of this paper. The mechanism is based on the assumption that loop corrections generate a singular dilaton kinetic function for a certain value of the dilaton field. For a nortgravitational source with a constant barotropic index, the system of cosmological equations reduces to an autonomous, third-order dynamical system. The behavior of the general solution in the vicinity of singularities of the dilaton coupling function is investigated by methods of the qualitative theory of dynamical systems for different values of the singularity index. The conditions under which solutions with a constant dilaton are attractors for a general solution with a variable dilaton are determined. The evolution of models is considered, the corresponding phase diagrams are constructed, and the question of the effectiveness of dilaton stabilization is investigated.