Fast-propagating waves in the phase of incommensurate structures, called phasons, have long been argued to enhance thermal transport. Although supersonic phason velocities have been observed, the lifetimes, from which mean free paths can be determined, have not been resolved. Using inelastic neutron scattering and thermal conductivity measurements, we establish that phasons in piezoelectric fresnoite make a major contribution to thermal conductivity by propagating with higher group velocities and longer mean free paths than phonons. The phason contribution to thermal conductivity is maximum near room temperature, where it is the single largest contributing degree of freedom.