In this paper, we examine a variant of the voter model on a dynamically changing network where agents have the option of changing their friends rather than changing their opinions. We analyse, in the context of dense random graphs, two models considered in Durrett et al. [Proc. Natl. Acad. Sci. USA 109 (2012) 3682–3687]. When an edge with two agents holding different opinion is updated, with probability $\frac{\beta }{n}$, one agent performs a voter model step and changes its opinion to copy the other, and with probability $1-\frac{\beta }{n}$, the edge between them is broken and reconnected to a new agent chosen randomly from (i) the whole network (rewire-to-random model) or, (ii) the agents having the same opinion (rewire-to-same model). We rigorously establish in both the models, the time for this dynamics to terminate exhibits a phase transition in the model parameter $\beta $. For $\beta $ sufficiently small, with high probability the network rapidly splits into two disconnected communities with opposing opinions, whereas for $\beta $ large enough the dynamics runs for longer and the density of opinion changes significantly before the process stops. In the rewire-to-random model, we show that a positive fraction of both opinions survive with high probability.