Joint work with Balázs Ráth. The voter model on \(\mathbb{Z}^d\) 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 \(\mu_{\alpha}\), for \(\alpha\) between 0 and 1. A configuration sampled from \(\mu_{\alpha}\) is a field of 0’s and 1’s on \(\mathbb{Z}^d\) in which the density of 1’s is \(\alpha\). We consider such a configuration from the point of view of site percolation on \(\mathbb{Z}^d\). We prove that in dimensions 5 and higher, the probability of existence of an infinite percolation cluster exhibits a phase transition in \(\alpha\). 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).