  We study the cosmological evolution of massless single-field scalar-tensor
theories of gravitation from the time before the onset of $e^+e^-$ annihilation
and nucleosynthesis up to the present. The cosmological evolution together with
the observational bounds on the abundances of the lightest elements (those
mostly produced in the early universe) place constraints on the coefficients of
the Taylor series expansion of $a(\phi)$, which specifies the coupling of the
scalar field to matter and is the only free function in the theory. In the case
when $a(\phi)$ has a minimum (i.e., when the theory evolves towards general
relativity) these constraints translate into a stronger limit on the
Post-Newtonian parameters $\gamma$ and $\beta$ than any other observational
test. Moreover, our bounds imply that, even at the epoch of annihilation and
nucleosynthesis, the evolution of the universe must be very close to that
predicted by general relativity if we do not want to over- or underproduce
$^{4}$He. Thus the amount of scalar field contribution to gravity is very small
even at such an early epoch.
