Summary: We obtain weak conditions to guarantee the Hyers-Ulam-Rassias stability of (nonlinear) Volterra integral equations with delay. In particular, this leads to a generalization of some results previously known. Basically, this is done by using certain weight functions in the framework of the space of continuous functions. Indeed, the method consists in a convenient combination of the classical Banach fixed point theorem together with a consideration of a weighted metric. Therefore, we avoid the use of the strict successive approximation method and also the consideration of generalized metrics (which need to be typically combined with a consequent fixed point alternative theorem). Some concrete examples are presented at the end of the paper.