A cooperative game with diversity constraints is given by a cooperative game, a coalition structure which partitions the set of players into communities, and a vector of integers specifying, for each community, the minimal number of its members that a coalition must possess to be considered as diverse. We provide axioms for a value on the class of such cooperative games with diversity constraints. Some combinations of axioms characterize two values inspired by the Shapley value (Shapley, 1953) and the Owen value (Owen, 1977) for games with a coalition structure. More specifically, the Diversity Owen value is characterized as the Owen value of the diversity-restricted game with a coalition structure, where the diversity-restricted game assigns a null worth to a coalition if it does not meet the diversity requirements or its original worth otherwise. Similarly, the Diversity Shapley value is characterized as the Shapley value of the diversity-restricted game (without coalition structure). Some of our axiomatic characterizations can be adapted to the class of simple games by replacing the Additivity axiom by the Transfer axiom (Dubey, 1975).