  We continue our study of the parabolic Anderson equation $\partial
u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t)$, $x\in\Z^d$, $t\geq
0$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the
discrete Laplacian, and $\xi$ plays the role of a \emph{dynamic random
environment} that drives the equation. The initial condition $u(x,0)=u_0(x)$,
$x\in\Z^d$, is taken to be non-negative and bounded. The solution of the
parabolic Anderson equation describes the evolution of a field of particles
performing independent simple random walks with binary branching: particles
jump at rate $2d\kappa$, split into two at rate $\xi \vee 0$, and die at rate
$(-\xi) \vee 0$. We assume that $\xi$ is stationary and ergodic under
translations in space and time, is not constant and satisfies
$\E(|\xi(0,0)|)<\infty$, where $\E$ denotes expectation w.r.t.\ $\xi$. Our main
object of interest is the quenched Lyapunov exponent $\lambda_0 (\kappa) =
\lim_{t\to\infty} \frac{1}{t}\log u(0,t)$. In earlier work we showed that under
certain mild space-time mixing assumptions the limit exists $\xi$-a.s., is
finite and continuous on $[0,\infty)$, is globally Lipschitz on $(0,\infty)$,
is not Lipschitz at 0, and satisfies $\lambda_0(0) = \E(\xi(0,0))$ and
$\lambda_0(\kappa) > \E(\xi(0,0))$ for $\kappa \in (0,\infty)$.In the present
paper we show that $\lim_{\kappa\to\infty} \lambda_0(\kappa) =\E(\xi(0,0))$
under an additional space-time mixing condition on $\xi$. This result shows
that the parabolic Anderson model exhibits space-time ergodicity in the limit
of large diffusivity. This fact is interesting because there are choices of
$\xi$ that fulfill our assumption for which the annealed Lyapunov exponent
$\lambda_1(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log \E(u(0,t))$ is infinite
on $[0,\infty)$, a situation that is referred to as strongly catalytic
behavior.
