We give conditions to obtain cosmological asymptotic freedom in scalar
-tensor theories of gravity. We show that this feature can be achieved
in FRW flat spacetimes since we obtain singularity free solutions whe
re the effective gravitational constant G(eff) --> 0 for t --> -infini
ty and, for some of them, G(eff) --> G(N) for t --> infinity, where G(
N) is the Newton constant.