We study the diffusion of a true and a false message (the rumor) in a social network. Upon hearing a message, individuals may believe it, disbelieve it, or debunk it through costly verification. Whenever the truth survives in steady state, so does the rumor. Communication intensity in itself is irrelevant for relative rumor prevalence, and the effect of homophily depends on the exact verification process and equilibrium verification rates. Our model highlights that successful policies in the fight against rumors increase individuals' incentives to verify.