Universal single-qubit operations based on purely geometric phase factors in adiabatic processes are demonstrated by utilizing a four-level system in a trapped single $^{40}$Ca$^+$ ion connected by three oscillating fields. Robustness against paramet
er variations is studied. The scheme demonstrated here can be employed as a building block for large-scale holonomic quantum computations, which may be useful for large qubit systems with statistical variations in system parameters.
