  In this paper we consider a model of scalar-tensor theory of gravitation in
which the scalar field, $\phi$ determines the gravitational coupling G and has
a Lagrangian of the form, $\mathcal{L}_{\phi} =-V(\phi)\sqrt{1 -
\partial_{\mu}\phi\partial^{\mu}\phi}$. We study the cosmological consequence
of this theory in the matter dominated era and show that this leads to a
transition from an initial decelerated expansion to an accelerated expansion
phase at the present epoch. Using observational constraints, we see that the
effective equation of state today for the scalar field turns out to be
$p_{\phi}=w_{\phi}{\rho}_{\phi}$, with $w_{\phi}=-0.88$ and that the transition
to an accelerated phase happened at a redshift of about 0.3.
