We have measured the temperature-dependent thermal conductivity kappa(T) of individual multiwall boron nitride nanotubes using a microfabricated test fixture that allows direct transmission electron microscopy characterization of the tube being measured. kappa(T) is exceptionally sensitive to isotopic substitution, with a 50% enhancement in kappa(T) resulting for boron nitride nanotubes with 99.5% 11B. For isotopically pure boron nitride nanotubes, kappa rivals that of carbon nanotubes of similar diameter.