We prove that on any surface there is a C^\infty diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoe \Lambda which is equivalent to the unique non-trivial Hausdorff measure in \Lambda. The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoe \Lambda are relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.