The authors studied the stability of a cold axially symmetric rotating plasma against axially symmetric perturbations. They demonstrate that a plasma of this kind can only be aperiodically unstable. A plasma with a rotation frequency Ω that is much smaller than the ratio of the mean Alfvén velocity A to the plasma radius a, and with a sufficiently smooth equilibrium parameter profile is unstable with negligibly small growth rates (much less than the rotation frequency Ω). At high rotation frequencies Ω (Ω ∼ A/a) the growth rate for the plasma instability γ can be made fairly small (γ ≪ Ω) by suitably selecting the radial dependence of the equilibrium parameters.