Stable distributions are of fundamental importance in probability theory, yet their absolute continuity makes them unsuitable for modeling count data. A discrete analog of strict stability has been previously proposed by replacing scaling with binomial thinning, but it only holds for a subset of the tail index parameters. Here, we generalize the discrete stable class to the full range of tail indices and show that it is equivalent to the mixed Poisson-stable family. This broadly discrete stable family is discretely infinitely divisible, with a compound Poisson representation involving a novel generalization of the Sibuya distribution. Under additional parameter constraints, they are also discretely self-decomposable and unimodal. The discrete stable distributions provide a new frontier in probabilistic modeling of both light and heavy tailed count data.