Using a variational procedure, within the effective mass-approximation, we calculate the binding energy of a shallow-donor impurity in a rectangular cross-sectional area of a GaAs quantum-well wire, under the action of an electric field applied perpendicular to one of the interfaces, assuming an infinite-confinement potential. We study the binding energy of the donor impurity as a function of the system geometry, the applied electric field, and the donor-impurity position. It was found that the presence of the electric field breaks down the degeneracy of the states for impurities symmetrically positioned within the structure. Our results for a large length of one side of the cross-sectional area coincide quite well with previous results in quantum wells. We show, unambiguously, that the impurity binding energy depends strongly not only on quantum confinement, but also on the applied electric field and on the distribution of impurities inside the quantum-well wires; these aspects must be taken into account in the quantitative understanding of optical phenomena related to shallow impurities when an electric field is applied.