The Danish stød, a kind of glottal prosody associated with certain syllables, as in barʔn ‘child’ (cf. stødless barnlig ‘childish’), has long been the target of intense phonological investigation. In this paper, we show that its analysis requires an understanding of the prosodic constituent structure of Danish, and of the essential role of the perfect prosodic word (coextensive with one foot). After motivating this notion on independent grounds, both in other languages and in the context of acquisition, we show that the Danish stød system, analyzed in Optimality Theory, provides a window on the workings of the perfect prosodic word, regulating the presence and absence of stød in some of the much-discussed cases in the literature. In conclusion, we discuss the status of the perfect prosodic word in the light of recent developments in phonological theory, such as Match Theory.