Joint work with Balázs Ráth. The voter model on is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When the model is considered in dimension 3 or higher, its set of (extremal) stationary distributions is equal to a family of measures , for between 0 and 1. A configuration sampled from is a field of 0's and 1's on in which the density of 1's is . We consider such a configuration from the point of view of site percolation on . We prove that in dimensions 5 and higher, the probability of existence of an infinite percolation cluster exhibits a phase transition in . If the voter model is allowed to have long range, we prove the same result for dimensions 3 and higher. These results partly settle a conjecture of Bricmont, Lebowitz and Maes (1987).