  We examine big bang nucleosynthesis (BBN) in models with a time-varying
gravitational constant $G$, when this time variation is rapid on the scale of
the expansion rate $H$, i.e, $\dot G/G \gg H$. Such models can arise naturally
in the context of scalar-tensor theories of gravity and result in additional
terms in the Friedman equation. We examine two representative models: a
step-function evolution for $G$ and a rapidly-oscillating $G$. In the former
case, the additional terms in the Friedman equation tend to cancel the effects
of an initial value of $G$ that differs from the present-day value. In the case
of deuterium, this effect is large enough to reverse the sign of the change in
(D/H) for a given change in the initial value of $G$. For rapidly-oscillating
$G$, the effect on the Friedman equation is similar to that of adding a vacuum
energy density, and BBN allows upper limits to be placed on the product of the
oscillation frequency and amplitude. The possibility that a rapidly oscillating
$G$ could mimic a cosmological constant is briefly discussed.
