We report the observation of a strong Fermi-edge singularity (FES), with the complete suppression of the band-edge peak, in the photoluminescence spectra of ultra-high-quality modulation-doped AlGaAs/GaAs quantum wires (QWRs). We find that the FES effect is very sensitive to the Fermi energy. The strong FES is observed only in QWRs having a Fermi energy of the order of a few meV, and disappears almost completely when the Fermi energy exceeds 10 meV. These results are expected to spark new research activities, both experimentally and theoretically, on many-body effects in one-dimensional electron gas.