  We study the transport properties of interacting electrons in a disordered
quantum wire within the framework of the Luttinger liquid model. We demonstrate
that the notion of weak localization is applicable to the strongly correlated
one-dimensional electron system. Two alternative approaches to the problem are
developed, both combining fermionic and bosonic treatment of the underlying
physics. We calculate the relevant dephasing rate, which for spinless electrons
is governed by the interplay of electron-electron interaction and disorder,
thus vanishing in the clean limit. Our approach provides a framework for a
systematic study of mesoscopic effects in strongly correlated electron systems.
