A homogeneous mixture of iso-butoxyethanol/water of noncritical composition is brought from a state in the thermodynamic stable region of its phase diagram in the vicinity of its lower critical point to a metastable state close to the binodal curve by a fast pressure jump. It reaches its thermodynamic equilibrium state by separating into two coexisting liquid phases of different compositions. The emerging second liquid phase is formed by nucleation and growth of droplets. After a short transition time, the early stages of the diffusion limited growth of the droplets of the emerging second liquid phase are studied by light scattering (Mie scattering). The experiments are carried out as function of the temperature difference (T-p - T-c) along the two branches of the liquid/liquid coexistence curve (T-c, critical temperature; T-p, temperature of phase separation). The mutual diffusion coefficient D extracted from the kinetics of the droplet growth decreases approaching the critical point along the two branches of the binodal curve. The values of D obtained in this way are in good agreement with those characterizing the dynamics of concentration fluctuations in each of the coexisting phases in thermodynamic equilibrium monitored by dynamic light scattering experiments.