Rapa Nui (also known as Easter Island) is one of the most isolated inhabited places in the world. It has captured the imagination of many owing to its archaeological record, which includes iconic megalithic statues called moai1. Two prominent contentions have arisen from the extensive study of Rapa Nui. First, the history of the Rapanui has been presented as a warning tale of resource overexploitation that would have culminated in a major population collapse—the ‘ecocide’ theory2–4. Second, the possibility of trans-Pacific voyages to the Americas pre-dating European contact is still debated5–7. Here, to address these questions, we reconstructed the genomic history of the Rapanui on the basis of 15 ancient Rapanui individuals that we radiocarbon dated (1670–1950 ce) and whole-genome sequenced (0.4–25.6×). We find that these individuals are Polynesian in origin and most closely related to present-day Rapanui, a finding that will contribute to repatriation efforts. Through effective population size reconstructions and extensive population genetics simulations, we reject a scenario involving a severe population bottleneck during the 1600s, as proposed by the ecocide theory. Furthermore, the ancient and present-day Rapanui carry similar proportions of Native American admixture (about 10%). Using a Bayesian approach integrating genetic and radiocarbon dates, we estimate that this admixture event occurred about 1250–1430 ce.