  We show that given an infinite triangulation $K$ of a surface with punctures
(i.e., with no vertices at the punctures) and a set of target cone angles
$\Theta$ at the punctures that satisfy a Gauss-Bonnet inequality, there exists
a hyperbolic metric that has the prescribed angles and supports a circle
packing in the combinatorics of $K$. Moreover, if $K$ is very symmetric, then
we can identify the underlying Riemann surface and show that it does not depend
on the angles. In particular, this provides examples of a triangulation $K$ and
a conformal class $X$ such that there are infinitely many conical hyperbolic
structures in the conformal class $X$ with a circle packing in the
combinatorics of $K$. This is in sharp contrast with a conjecture of
Kojima-Mizushima-Tan in the closed case.
