  A lattice regularized Lax operator for the nonultralocal modified Korteweg de
Vries (mKdV) equation is proposed at the quantum level with the basic operators
satisfying a $q$-deformed braided algebra. Finding further the associated
quantum $R$ and $Z$-matrices the exact integrability of the model is proved
through the braided quantum Yang--Baxter equation, a suitably generalized
equation for the nonultralocal models. Using the algebraic Bethe ansatz the
eigenvalue problem of the quantum mKdV model is exactly solved and its
connection with the spin-$\ha$ XXZ chain is established, facilitating the
investigation of the corresponding conformal properties.
