We measure the photoluminescence spectra for an array of modulation doped, T-shaped quantum wires as a function of the one-dimensional density n(e), which is modulated with a surface gate. We present self-consistent electronic structure calculations for this device which show a band-gap renormalization which, when corrected for excitonic energy and its screening, are largely insensitive to n(e) and which are in quantitatively excellent agreement with the data. The calculations show that electron and hole remain bound up to similar to3x10(6) cm(-1) and that, therefore, the stability of the exciton far exceeds the conservative Mott criterion, as determined from the Bethe-Salpeter equation.