We report a calculation of the binding and transition energies of the ground and some excited states of a hydrogenic donor impurity located at the axis of a cylindrical GaAs quantum-well wire, under the action of a magnetic field applied in the axial direction. Calculations are made using the effective-mass approximation within the variational approach for infinite confinement potential. Our results are obtained for several wire radii and as a function of the applied magnetic field. We have found that some excited states are not bounded for some values of the radius of the wire and of the applied magnetic field. We show how the geometric confinement and the applied magnetic field split the degeneracy of some excited states. Also, we compare our results with those found in GaAs-(Ga, Al)As quantum wells.