  We study the asymptotic behaviour of isotropic and homogeneous universes in
general scalar-tensor gravity theories containing a p=-rho vacuum fluid stress
and other sub-dominant matter stresses. It is shown that in order for there to
be approach to a de Sitter spacetime at large 4-volumes the coupling function,
omega(phi), which defines the scalar-tensor theory, must diverge faster than
|phi_infty-phi|^(-1+epsilon) for all epsilon>0 as phi rightarrow phi_infty <> 0
for large values of the time. Thus, for a given theory, specified by
omega(phi), there must exist some phi_infty in (0,infty) such that omega ->
infty and omega' / omega^(2+epsilon) -> 0 as phi -> 0 phi_infty in order for
cosmological solutions of the theory to approach de Sitter expansion at late
times. We also classify the possible asymptotic time variations of the
gravitation `constant' G(t) at late times in scalar-tensor theories. We show
that (unlike in general relativity) the problem of a profusion of ``Boltzmann
brains'' at late cosmological times can be avoided in scalar-tensor theories,
including Brans-Dicke theory, in which phi -> infty and omega ~ o(\phi^(1/2))
at asymptotically late times.
