  Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d
\geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r
\geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with
finite $(2d-2)$-nd moment (or if $d=2$, a finite $(2 + \varepsilon)$-moment
condition for some $\varepsilon >0$). Let $A \subset {\bf R}^d$ be compact with
a nice boundary. Let $\alpha $ be the expected volume of a ball of radius $Y$,
and suppose $r=r(t)$ is chosen so that $\alpha t r^d - \log t - (d-1) \log \log
t$ is a constant independent of $t$. A classical result of Hall and of Janson
determines the (non-trivial) large-$t$ limit of the probability that $A$ is
fully covered by $Z$. In this paper we provide an $O((\log \log t)/\log t)$
bound on the rate of convergence in that result. With a slight adjustment to
$r(t)$, this can be improved to $O(1/\log t)$.
