We characterise how the assimilation patterns of minorities into the strong and the weak language differ in a situation of asymmetric bilingualism. Using large variations in language composition in Canadian cities from the 2001 and 2006 Censuses, we show that the differences in the knowledge of English by immigrant allophones (i.e. the immigrants with a mother tongue other than English and French) in English-majority cities are mainly due to sorting across cities. Instead, in French-majority cities, learning plays an important role in explaining differences in knowledge of French. In addition, the presence of large anglophone minorities deters much more the assimilation into French than the presence of francophone minorities deters the assimilation into English. Finally, we find that language distance plays a much more important role in explaining assimilation into French, and that assimilation into French is much more sensitive to individual characteristics than assimilation into English. Some of these asymmetric assimilation patterns extend to anglophone and francophone immigrants, but no evidence of learning is found in this case.