  This is the second part of our result on a class of global characteristic
problems for the Einstein vacuum equations with small initial data. In the
previous work denoted by (I), our attention was focused on prescribing the
initial data satisfying the costraints imposed by the characteristic problem.
Here we show how the global existence result can be achieved. This part is
heavily based on the global results of D.Christodoulou, S.Klainerman and
S.Klainerman, F.Nicolo`.
