Sympathetic cooling of trapped ions has been demonstrated in an experiment where $^{198}\mathrm{Hg}^{+}$ ions were confined in a Penning ion trap with laser-cooled $^{9}\mathrm{Be}^{+}$ ions. $^{198}\mathrm{Hg}^{+}$ temperatures below 1 K were achieved. Ion plasma sizes, shapes, and rotation frequencies were measured. Dramatic changes in the $^{9}\mathrm{Be}^{+}$ plasma were observed when the $^{198}\mathrm{Hg}^{+}$ ions were introduced into the trap. These observations are consistent with the prediction of centrifugal separation for ions of different charge-to-mass ratios.