The mergers of black-hole neutron star binaries are calculated using a pseudo-general relativistic potential that incorporates post-Newtonian corrections. Both normal matter neutron stars and self-bound strange quark matter stars are considered as black-hole partners. As long as the neutron stars are not too massive relative to the black-hole mass, orbital decay terminates in stable mass transfer rather than an actual merger. For a normal neutron star, mass transfer results in a widening of the orbit but the stable transfer ends before the minimum neutron star mass is reached. For a strange star, mass transfer does not result in an appreciable enlargement of the orbital separation, and the stable transfer continues until the strange star essentially disappears. These differences might be observable through their respective gravitational wave signatures.