  Given a weighted line arrangement in the projective plane, with weights
satisfying natural constraint conditions, we show the existence of a Ricci-flat
K\"ahler metric with cone singularities along the lines asymptotic to a
polyhedral K\"ahler cone at each multiple point. Moreover, we discuss a
Chern-Weil formula that expresses the energy of the metric as a `logarithmic'
Euler characteristic with points weighted according to the volume density of
the metric.
