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Abstract. We consider a billiard in the punctured torus obtained by removing a small disk of radius ε > 0 from the flat torus T 2 , with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity ω only. Letτε(ω), and respectivelyRε(ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when T 2 is being identified with [0, 1) 2 . We prove that the probability measures on [0, ∞) associated with the random variables ετε and εRε are weakly convergent as ε → 0 + and explicitly compute the densities of the limits.