We have investigated the stellar kinematics of two brightest cluster galaxies (BCGs) to large distances from the cluster centre. In both galaxies, the velocity dispersion rises with radius. Orbital anisotropy is ruled out as the cause and we conclude that the kinematics in the outer parts of BCGs are dark matter dominated. This result contrasts with the current problem for intermediate luminosity ellipticals, which appear to have less dark matter than predicted by CDM. Combining the present results with our previous work, the fraction of BCGs with outwardly rising dispersions ($\sim60\%$) is much higher than that found previously ($\sim7\%$) from the inner parts of BCGs. We argue that this phenomenon may be more common than previously thought. The agreement with radial trends expected for a smooth galaxy/cluster transition, and the lack of dependence on cD/BCG dominance and cluster environment, suggest that the outwardly rising dispersion is a genuine response of the stars to the underlying cluster potential. There is evidence that BCG in Abell 2063 is not at the centre of the cluster potential.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html