Consider a sample of apparently unrelated individuals, for which marker genotype and phenotype data is available. When individuals are sampled on phenotypes, we propose an ascertained ancestral recombination graph (ARG) that models shared ancestry of the sample chromosomes given phenotype data along a region that possibly harbors a disease susceptibility gene. The ascertained ARG is used to define a gene mapping algorithm by means of a lod score and associated p-values based on permutation testing. Under certain modeling simplifications, the lod score and p-values can be computed exactly, without any Monte Carlo approximations, even for unphased chromosome genotype data. Our method handles incomplete penetrance, varying marker allele frequencies and neutral mutations, and is based on a Hidden Markov algorithm for subsets of disease mutated chromosomes. The performance of the method is investigated in a simulation study and for a real data set from a case-control study of breast cancer.