In Bunimovich and Del Magno [Semi-focusing billiards: hyperbolicity. Comm. Math. Phys.262 (2006), 17–32], we proved that billiards in certain three-dimensional convex domains are hyperbolic. In this paper, we continue the study of these systems, and prove that they enjoy the Bernoulli property. This result answers affirmatively a long-standing question on the existence of ergodic billiards in convex domains in dimensions greater than two. Besides, it shows that the chaotic components of the first rigorously investigated three-dimensional billiards with mixed phase space (mushroom billiards), introduced in Bunimovich and Del Magno, are in fact Bernoulli.