An exact solution is presented for the time-dependent wave function of an initial ground-state Kramers-doublet qubit that is driven around a quantum ring. We show that the initial qubit may be transformed to an arbitrary point on the Bloch sphere for an integral number of revolutions around the ring. Full coverage of the Bloch sphere is achieved by dividing the total rotation into segments, changing the rotation axis after each segment by an adiabatic shift in the Rashba spin-orbit interaction. Prospects and challenges for possible realizations are discussed for which rings based on InAs quantum wires are promising candidates.