The influence of diffusion on the kinetics of ligand binding to a macromolecule with two sites is considered for a simple model where, in the reaction-controlled limit, there is no cooperativity and hence the sites are independent. By applying our recently developed formalism to describe a network of coupled diffusion-influenced reactions, we show that the rate constants of chemical kinetics cannot just be renormalized. Rather a new reaction channel, which connects the two singly occupied states, must be introduced. The rate constants of this new channel depend on the committor or capture probability that a ligand that just dissociated from one site rebinds to the other. This result is rederived in an elementary way using the encounter complex model. Illustrative calculations are presented where the kinetics of the fractional saturation of one site is compared with that of a macromolecule that has only this site. If all sites are initially empty, then the second site slows down binding to the first due to competition between the sites. On the other hand, if the second site is initially occupied, the binding of the first site speeds up because of the direct diffusion-induced transitions between the two singly bound states.