In the present work, we analyze the stability properties of a one dimensional thermoelastic laminated beam with structural damping, where the heat conduction is given by Gurtin–Pipkin's law. We show that the associated solution semigroup is exponentially stable if and only if the new introduced stability number χ is null. Furthermore, if $\chi \ne 0$ we prove the lack of exponential stability. The proofs are based on the energy method and Gearhart–Herbst–Prüss–Huang theorem.