  The spin relaxation rate $1/\tau_s (W)$ in disordered quantum wires with
Rashba and Dresselhaus spin-orbit coupling is derived analytically as a
function of wire width $W$. It is found to be diminished when $W$ is smaller
than the bulk spin-orbit length $L_{\rm SO}$. Only a small spin relaxation rate
due to cubic Dresselhaus coupling $\gamma$ is found to remain in this limit. As
a result, when reducing the wire width $W$ the quantum conductivity correction
changes from weak anti- to weak localization and from negative to positive
magnetoconductivity.
