Results for the binding and infrared transition energies associated with the ground and first few excited states of a hydrogenic-donor impurity in a cylindrical GaAs-${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As quantum-well wire are presented. A variational approach for the trial envelope wave functions corresponding to each donor state is used within the effective-mass approximation. Results are obtained for several wire radii in the case of donors located at the axis of the quantum-well wire. We find that the impurity states 2${\mathit{p}}_{\mathit{x}}$ (2${\mathit{p}}_{\mathit{y}}$) and 3${\mathit{p}}_{\mathit{x}}$ (3${\mathit{p}}_{\mathit{y}}$) are not bounded when the wire radius is sufficiently small, a behavior similar to the one found for impurities in quantum-well heterostructures.