In this paper, we establish an existence theorem for a cubic Urysohn-Stieltjes integral equation in the Banach space $C([0,1])$. The equation under consideration is a general form of numerous integral equations encountered in the theory of radioactive transfer, in the kinetic theory of gases and in the theory of neutron transport. Our main tools are the measure of noncompactness (related to monotonicity) and a fixed point theorem due to Darbo.