  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 where the effective
gravitational constant $G_{eff}\rightarrow 0$ for $t\rightarrow -\infty$ and,
for some of them, $G_{eff}\rightarrow G_{N}$ for $t\rightarrow\infty$, where
$G_{N}$ is the Newton constant.
