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The density of states for a chaotic billiard with randomly distributedpoint-like scatterers is calculated, doubly averaged over the positions of theimpurities and the shape of the billiard. Truncating the billiard Hamiltonianto a N x N matrix, an explicit analytic expression is obtained for the case ofbroken time-reversal symmetry, depending on rank N of the matrix, number L ofscatterers, and strength of the scattering potential. In the strong couplinglimit a discontinuous change is observed in the density of states as soon as Lexceeds N

A supersymmetry approach to billiards with randomly distributed scatterers