The kinetics of spinodal decomposition have been investigated in a one-dimensional kinetic Ising model by Monte Carlo simulation techniques. Three kinetic regimes can be clearly identified following relatively deep quenches from a random state. An early-time regime characterized by temperature-independent growth of the average domain size evolves to a ${t}^{\frac{1}{3}}$ growth regime via an incubation period. Single-atom diffusion is shown to be important to the growth process only in the earliest regime. The scaling properties of the domain-size distribution indicate that the onset of the dynamic scaling occurs after the final growth regime has begun.