  The quantum theory of the Liouville model with imaginary field is considered
using the quantum inverse scattering method. An integrable structure with
nontrivial spectral parameter dependence is developed for lattice Liouville
theory by scaling the $L$-matrix of lattice sine-Gordon theory. This $L$-matrix
yields Bethe Ansatz equations for Liouville theory, by the methods of the
algebraic Bethe Ansatz. Using the string picture of exited Bethe states, the
lattice Liouville Bethe equations are mapped to the corresponding equations for
the spin 1/2 XXZ chain. The well developed theory of finite size corrections in
spin chains is used to deduce the conformal properties of the lattice Liouville
Bethe states. The unitary series of conformal field theories emerge for
Liouville couplings of the form $\gam = \pi\frac{\nu}{\nu+1}$, corresponding to
root of unity XXZ anisotropies. The Bethe states give the full spectrum of the
corresponding unitary conformal field theory, with the primary states in the
\Kac table parameterized by a string length $K$, and the remnant of the chain
length mod $(\nu+1)$.
