An understanding of the topological structure of the Internet is needed for quite a number of networking tasks, e. g., making decisions about peering relationships, choice of upstream providers, inter-domain traffic engineering. One essential component of these tasks is the ability to predict routes in the Internet. However, the Internet is composed of a large number of independent autonomous systems (ASes) resulting in complex interactions, and until now no model of the Internet has succeeded in producing predictions of acceptable accuracy.We demonstrate that there are two limitations of prior models: (i) they have all assumed that an Autonomous System (AS) is an atomic structure - it is not, and (ii) models have tended to oversimplify the relationships between ASes. Our approach uses multiple quasi-routers to capture route diversity within the ASes, and is deliberately agnostic regarding the types of relationships between ASes. The resulting model ensures that its routing is consistent with the observed routes. Exploiting a large number of observation points, we show that our model provides accurate predictions for unobserved routes, a first step towards developing structural mod-els of the Internet that enable real applications.