  We consider the Poisson Boolean percolation model in $\mathbb{R}^2$, where
the radii of each ball is independently chosen according to some probability
measure with finite second moment. For this model, we show that the two
thresholds, for the existence of an unbounded occupied and an unbounded vacant
component, coincide. This complements a recent study of the sharpness of the
phase transition in Poisson Boolean percolation by the same authors. As a
corollary it follows that for Poisson Boolean percolation in $\mathbb{R}^d$,
for any $d\ge2$, finite moment of order $d$ is both necessary and sufficient
for the existence of a nontrivial phase transition for the vacant set.
