We argue that for ballistic transport through a narrow conductor (of width $W$) a threshold magnetic field exists below which the Hall resistance vanishes. The field is of order $(\frac{h}{e})k_{\mathrm{F}}^{}{}_{}{}^{\ensuremath{-}1}{W}^{\ensuremath{-}3}$, and is reached when the transverse wavelength of quantum edge states becomes comparable to the width. This is offered as a mechanism for the quenching of the Hall effect discovered experimentally in a narrow two-dimensional electron-gas wire by Roukes et al.