The organisation of who speaks when in conversation is perhaps the most fundamental aspect of human communication. Research on a wide variety of groups of speakers has revealed a seemingly universal preference for between-speaker transitions consisting of very short silent gaps. Previous research on conversational turn-taking in Autism Spectrum Disorder (ASD) consists of only a handful of studies, most of which are limited in scope and based on the non-spontaneous speech of children and adolescents. No previous studies have investigated dialogues between autistic adults. We analysed the conversational turn-taking behaviour of 28 adult native German speakers in two groups of dyads, in which both interlocutors either did or did not have a diagnosis of ASD. We found no clear difference in turn-timing between the ASD and the control group overall, with both groups showing the same preference for very short silent-gap transitions that has been described for many other groups of speakers in the past. We did, however, find a clear difference between groups specifically in the earliest stages of dialogue, where ASD dyads produced considerably longer silent gaps than controls. We discuss our findings in the context of the previous literature, the implications of diverging behaviour specifically in the early stages of conversation, and the general importance of studying the neglected aspect of interactions between autistic adults.