A spatially discrete model for phase separation with conserved order parameter is proposed. This one-dimensional model is obtained as the deterministic limit of an anisotropic lattice gas. A particular choice is made for the jump rates (which still fulfill detailed balance conditions) so that the resulting model is mathematically tractable. It exhibits a phase transition of first-order type whose nonlinear dynamics is investigated using both analytical and numerical methods. All the stationary solutions with zero current are found and parametrized in terms of Jacobian elliptic functions, showing a striking similarity with the nonlinear (continuous) Cahn-Hilliard equation. In the limit of infinite wavelength, particular solutions are found which descrive isolated domains of arbitrary size embedded in an homogeneous infinite medium of the opposite phase. New results are also presented on the structure of the set of solutions. Time-dependent profiles are studied in the spinodal regime and the stability of bounded stationary solutions is also investigated in this context. A description of time-dependent profiles is proposed which considers only interactions between neighboring domains and makes use of isolated domain solutions. This approach results in an analytic expression for the exponents characteristic of the instability of stationary solutions and is validated by comparison to numerical values. Qualitative results are also discussed and the relation to the Cahn-Hilliard equation is emphasized.