Recent work has leveraged the unique genealogical structure and extensive genotyping (>30%) of the Icelandic population to perform long-range phasing (LRP), enabling accurate imputation and association analysis of rare variants in target samples typed on genotyping arrays. Here, we develop a fast and accurate LRP method, Eagle, that extends this paradigm to outbred populations by harnessing long (>4cM) identical-by-descent (IBD) tracts shared among distantly related individuals. We applied Eagle to N=150K samples (0.2% of the British population) from the UK Biobank, and we determined that it is 1-2 orders of magnitude faster than existing methods while achieving exquisite phasing accuracy (switch error rate ≈0.3%, corresponding to perfect phase at the scale of >10Mb). Moreover, we observed that Eagle imputed masked genotypes with accuracy R2>0.75 down to a minor allele frequency of 0.1%. Compared to computationally tractable alternatives, Eagle attained large improvements in phasing and imputation accuracy at N=150K and smaller improvements at smaller sample sizes, illustrating the advantages that LRP-based imputation will yield as very large reference panels become available.