N. Chernov and S. Troubetzkoy Ergodicity of Billiards in Polygons with Pockets (28K, LATeX) ABSTRACT. The billiard in a polygon is not always ergodic and never K-mixing or Bernoulli. Here we consider billiard tables by attaching disks to each vertex of an arbitrary simply connected, convex polygon. We show that the billiard on such a table is ergodic, K-mixing and Bernoulli.