  We prove that given a fixed radius $r$, the set of isometry-invariant
probability measures supported on ``periodic'' radius $r$-circle packings of
the hyperbolic plane is dense in the space of all isometry-invariant
probability measures on the space of radius $r$-circle packings. By a periodic
packing, we mean one with cofinite symmetry group. As a corollary, we prove the
maximum density achieved by isometry-invariant probability measures on a space
of radius $r$-packings of the hyperbolic plane is the supremum of densities of
periodic packings. We also show that the maximum density function varies
continuously with radius.
