In a founder population patients with a genetic disease are likely to share predisposing genes from a common ancestor. We show that, depending on the distance of the relationship, patients are expected to share extended segments of DNA around the disease gene. Because of the size of the shared segment, a genomic search with DNA markers for such shared segments, identity by descent (IBD) mapping, can efficiently find the map position of genes, particularly due to genetic drift leading to reduction of heterogeneity and the large number of meioses that is implicitly observed. The statistical power of this method and the approximate cost are given as a function of the density of the map of tested markers and the number of generations since a common ancestor. Initial marker spacings between 5 and 15 centiMorgans are shown to be optimal. IBD mapping is applicable to many genetic diseases, because it does not presuppose a specific genetic model.