We consider the dynamical system of Sinai billiards with a cusp where two walls of the billiard table meet at the vertex of a cusp and have zero one-sided curvature, thus forming a "flat point" at the vertex. For Holder continuous observables (random variables), we show that the properly normalized Birkhoff sums of stationary variables, with respect to the so-called ergodic billiard map, converge in distribution to a totally skewed alpha-stable law, for some alpha between 1 and 2.