This study reports on findings from an experimental investigation into the knowledge of binding (the module of grammar regulating the distribution of reflexives and pronouns), in a group of young adults with Down syndrome (DS). Participants with DS were found to have difficulties comprehending reflexives, but not pronouns. In contrast with pronouns, which are interpreted by invoking extra-syntactic mechanisms, the interpretation of reflexives depends on a syntactic relation between the reflexive element and its antecedent. This points to the deficit in DS being syntactic in nature. Such a pattern is exactly the opposite to that found in typically developing English children, who obey the syntactic constraints on the distribution of reflexives early on, but have trouble applying the co-reference rule, a constraint outside syntax proper that regulates the interpretation of pronouns. This result provides evidence that language in DS is not merely delayed, as traditionally described, but also deficient in important respects: the deficit amounts to an inability to establish the syntactic relation between the anaphor and its antecedent. The revealed grammatical deficit supports the hypothesis that modules of the computational system, such as morphosyntax, are relatively more impaired in this population, than those associated with the general processing system, such as lexical knowledge or pragmatics.