This paper is based on bilingual conversations we have collected in two bilingual settings, namely service encounters in Barcelona and ordinary informal conversations among bilingual Rwandese in Belgium. On the basis of these data, the paper raises the question of whether the concept of language is a useful one in accounting for language choice acts among bilingual speakers. Observation of the data reveals that the concept of language can account for bilinguals' language choice acts only partially. This observation is found to be consistent with Gumperz' (1982) argument according to which, in talk, speakers themselves do not necessarily orient to the notion of language, but rather to their â??own notion of code.â?� In the paper, we refer to that speakers' own code as the medium of a bilingual conversation. Therefore, our main argumentis that, for an account of language choice among bilingual speakers, the concept of language need be suspended in favor of that of the medium of a bilingual conversation. In the paper, three connected questions are specifically addressed. How do speakers themselves establish the medium? What versions may the medium take? How can analysts discover the medium?