We introduce a conceptual bridge between the previously unlinked fields of phylogenetics and mathematical spatial ecology, which enables the spatial parameters of an emerging epidemic to be directly estimated from sampled pathogen genome sequences. By using phylogenetic history to correct for spatial autocorrelation, we illustrate how a fundamental spatial variable, the diffusion coefficient, can be estimated using robust nonparametric statistics, and how heterogeneity in dispersal can be readily quantified. We apply this framework to the spread of the West Nile virus across North America, an important recent instance of spatial invasion by an emerging infectious disease. We demonstrate that the dispersal of West Nile virus is greater and far more variable than previously measured, such that its dissemination was critically determined by rare, long-range movements that are unlikely to be discerned during field observations. Our results indicate that, by ignoring this heterogeneity, previous models of the epidemic have substantially overestimated its basic reproductive number. More generally, our approach demonstrates that easily obtainable genetic data can be used to measure the spatial dynamics of natural populations that are otherwise difficult or costly to quantify.