Black hole–neutron star ( BHNS ) binaries are important sources of gravitational waves for second-generation interferometers , and BHNS mergers are also a proposed engine for short , hard gamma-ray bursts . The behavior of both the spacetime ( and thus the emitted gravitational waves ) and the neutron star matter in a BHNS merger depend strongly and nonlinearly on the black hole ’ s spin . While there is a significant possibility that astrophysical black holes could have spins that are nearly extremal ( i.e . near the theoretical maximum ) , to date fully relativistic simulations of BHNS binaries have included black-hole spins only up to S / M ^ { 2 } =0.9 , which corresponds to the black hole having approximately half as much rotational energy as possible , given the black hole ’ s mass . In this paper , we present a new simulation of a BHNS binary with a mass ratio q = 3 and black-hole spin S / M ^ { 2 } =0.97 , the highest simulated to date . We find that the black hole ’ s large spin leads to the most massive accretion disk and the largest tidal tail outflow of any fully relativistic BHNS simulations to date , even exceeding the results implied by extrapolating results from simulations with lower black-hole spin . The disk appears to be remarkably stable . We also find that the high black-hole spin persists until shortly before the time of merger ; afterwards , both merger and accretion spin down the black hole .