We present new redshifts of ten rich clusters of galaxies of the Abell, Corwin, and Olowin cluster catalog (Abell's southern extension). These are the first results of a survey of southern cluster redshifts which we intend to complete out to z ≃ 0.1. In six clusters we measured the redshifts of the first and second brightest galaxy, obtaining typical differences of ≃650 km s-1. This gives confidence in both the identification of the brightest galaxies and the redshift measurement. We also found that the redshifts we obtained agree well with the log (cz)-m10 relationship.