The ability to generate new meaning by rearranging combinations of meaningless sounds is a fundamental component of language. Although animal vocalizations often comprise combinations of meaningless acoustic elements, evidence that rearranging such combinations generates functionally distinct meaning is lacking. Here, we provide evidence for this basic ability in calls of the chestnut-crowned babbler (Pomatostomus ruficeps), a highly cooperative bird of the Australian arid zone. Using acoustic analyses, natural observations, and a series of controlled playback experiments, we demonstrate that this species uses the same acoustic elements (A and B) in different arrangements (AB or BAB) to create two functionally distinct vocalizations. Specifically, the addition or omission of a contextually meaningless acoustic element at a single position generates a phoneme-like contrast that is sufficient to distinguish the meaning between the two calls. Our results indicate that the capacity to rearrange meaningless sounds in order to create new signals occurs outside of humans. We suggest that phonemic contrasts represent a rudimentary form of phoneme structure and a potential early step towards the generative phonemic system of human language.