Abstract. This paper is devoted to determining the scalar relaxation kernel a in a second-order (in time) integrodifferential equation related to a Banach space when an additional measurement involving the state function is available. A result concerning global existence and uniqueness is proved. The novelty of this paper consists in looking for the kernel a in the Banach space BV (0, T), consisting of functions of bounded variations, instead of the space W1,1(0, T) used up to now to identify a. An application is given, in the framework of L2-spaces, to the case of hy- perbolic second-order integrodifferential equations endowed with initial and Dirichlet boundary conditions.