  We study the Bergman space interpolation problem of open Riemann surfaces
obtained from a compact Riemann surface by removing a finite number of points.
We equip such a surface with what we call an asymptotically flat conformal
metric, i.e., a complete metric with zero curvature outside a compact subset.
We then establish necessary and sufficient conditions for interpolation in
weighted Bergman spaces over asymptotically flat Riemann surfaces.
