We present a new method for controlling vortex breakdown (VB) via addition of co- or counter-rotation near the axis. Co-rotation is adequate to totally suppress VB, whereas counter-rotation increases the number and size of VB "bubbles" and makes the flow unsteady. We study these effects in a closed cylindrical container, in which a rotating end disc drives the base flow; an independently rotating central rod (with rod radius≪disk radius) is employed to control VB. This flow, being free of ambient disturbances, is well suited for understanding both the VB mechanism and its control; the present work appears to be the first to study VB control. We develop and explain our control strategy using flow visualization and simple analytical reasoning. Our results suggest that an additional co- or counter-rotation, applied near the vortex axis, can be effective in suppressing or enhancing VB in practical flows.