This paper describes the occurrence and development of double-diffusive convection in the liquid phase during solidification of NH4Cl-H2O system in a confined cavity with lateral cooling. Multiple point measurements of concentration show a step change in the vertical direction, and temperature visualizations reveal an S-shaped profile of isotherms, which indicate the existence of time-dependent horizontally-stacked roll cells separated by diffusive interfaces. The cells are generated in a sequential fashion, rather than simultaneously and the thickness of each cell increases with progression of solidification. The concentration in each cell is found to remain nearly constant, except for the initial development of cells. Convection within each cell is largely controlled by the temperature field, and diffusion is dominant in the diffusive interface between two cells due to the solute field with a vertical concentration gradient. The fluid in the diffusive interface is initially stagnant, but the interface changes into a new cell under a certain condition. The criteria for cell generation are determined by the buoyancy ratio and the thermal Rayleigh number in the diffusive interface.