  We explore a version of the cosmological dilaton-fixing and decoupling
mechanism in which the dilaton-dependence of the low-energy effective action is
extremized for infinitely large values of the bare string coupling $g_s^2 =
e^{\phi}$. We study the efficiency with which the dilaton $\phi$ runs away
towards its ``fixed point'' at infinity during a primordial inflationary stage,
and thereby approximately decouples from matter. The residual dilaton couplings
are found to be related to the amplitude of the density fluctuations generated
during inflation. For the simplest inflationary potential, $V (\chi) = {1/2}
m_{\chi}^2 (\phi) \chi^2$, the residual dilaton couplings are shown to predict
violations of the universality of gravitational acceleration near the $\Delta a
/ a \sim 10^{-12}$ level. This suggests that a modest improvement in the
precision of equivalence principle tests might be able to detect the effect of
such a runaway dilaton. Under some assumptions about the coupling of the
dilaton to dark matter and/or dark energy, the expected time-variation of
natural ``constants'' (in particular of the fine-structure constant) might also
be large enough to be within reach of improved experimental or observational
data.
