  We study cosmology in scalar-tensor (Bergmann Wagoner) gravity, restricting
the coupling function, ${\omega}({\phi})$ to be constant. Rather than specify
the form of the cosmological function, ${\lambda}({\phi})$, the scalar field is
modelled as a function which decays to its present value ${\phi}=1$. Solutions
of the field equations are found for which ${\lambda}({\phi})$ evolves from a
large value (approximately 1) near the singularity to a small, non-zero value
at later times, avoiding the problem of pre-inflationary collapse in standard
general relativistic cosmology. Interpreting the model within the framework of
Brans-Dicke theory with a scalar potential, we find that for suitable initial
conditions, the reconstructed potential at late times for flat, open and closed
universes is well described by a Higgs-like Mexican hat potential, quartic in
${\phi}$, though this was not built in to the initial assumptions.
