The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini  (three infinite memories), Guesmia and Kirane  (two infinite memories), Guesmia  (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al.  (one infinite memory acting on the shear angle displacement). When the kernel functions have an exponential decay at infinity, the obtained stability estimates in these papers lead to the exponential stability of the system if the speeds ofwave propagations are the same, and to the polynomial one with decay rate [...] otherwise. The subject of this paper is to study the case where only one infinite memory is considered and it is acting on the vertical displacement. As far as we know, this case has never studied before in the literature. We show that this case is deeply different from the previous ones cited above by proving that the exponential stability does not hold even if the speeds of wave propagations are the same and the kernel function has an exponential decay at infinity. Moreover, we prove that the system is still stable at least polynomially where the decay rate depends on the smoothness of the initial data. For classical solutions, this decay rate is arbitrarily close to [...] . The proof is based on a combination of the energy method and the frequency domain approach to overcome the new mathematical difficulties generated by our system.