  We consider the Poisson cylinder model in ${\mathbb R}^d$, $d\ge 3$. We show
that given any two cylinders ${\mathfrak c}_1$ and ${\mathfrak c}_2$ in the
process, there is a sequence of at most $d-2$ other cylinders creating a
connection between ${\mathfrak c}_1$ and ${\mathfrak c}_2$. In particular, this
shows that the union of the cylinders is a connected set, answering a question
appearing in a previous paper. We also show that there are cylinders in the
process that are not connected by a sequence of at most $d-3$ other cylinders.
Thus, the diameter of the cluster of cylinders equals $d-2$.
