Despite their apparent simplicity, the structure of DPs containing “complement” CPs (what we will call DCs) has long been obscure. One major strand of investigation has attempted to assimilate DCs to (close) nominal apposition, implying that N and CP form a structural unit that then combines with D. Danish has two kinds of DCs, a bare DC with the superficial structure [D N CP] and a prepositional DC in which the CP is encased in a PP. Exploiting clues provided by the allomorphy of the definite morpheme, we argue that the bare and prepositional DCs have very different structures, neither of which can be assimilated to apposition between N and CP. We further show that the two kinds of DC have distinct semantic and pragmatic properties. We then argue that English also has two different structures for DCs, and that they are plausibly parallel to the structures we establish for Danish. We conclude by arguing that two distinct structures give rise to the “apposition” relation: in one case it is between coarguments of D and in the other it is nonrestrictive composition.