In this paper we obtain uniform estimates for the lattice point problem in the hyperbolic plane H under the assumption that the action is by a Fuchsian group Γ which is co-finite. We fix a point w from H and set Nt(z, w) equal to the number of translates of w by the group Γ which lie in a geodesic ball of radius t centred at a point z of H. The behaviour of Nt(z, w) is then examined when t is large and z is allowed to vary over H. We show that the finite quantity depends crucially on the point w, and indeed can become arbitrarily large with w. On the other hand, for the average of this quotient we derive the estimate as t → ∞, where the implied constant is an explicit function of w. In this formula, vol(F) is the hyperbolic volume of a Dirichletfundamental domain F for Γ, and ∣Γw∣ denotes the number of elements from Γ fixing w. This estimate is then combined with a recent sampling theorem of K. Seip to obtain an inequality which decides whether or not the orbit Γ . w forms a set of interpolation for a given weighted Bergman space in H. 1991 Mathematics Subject Classification: 11F72, 11P21, 30D35, 30E05, 30F35.