We study an inverse problem for the fractional wave equation with a potential, and the mathematical model for the fractional wave equation is formulated as follows. Let \(\Omega \subset {\mathbb R}^n\) be a non-empty bounded Lipschitz domain, for \(n \in {\mathbb N}\) . Given \(T>0\) , \( s \in (0,1)\) and \(q=q(x)\in L^\infty (\Omega ) \) , consider the initial exterior value problem for the wave equation with the fractional Laplacian, \(\displaystyle \begin{aligned} \begin {cases} \left ( \partial _t^2 + (-\Delta )^s + q \right ) u =0 & \text{ in }\Omega \times (0,T) ,\\ u=f & \text{ in } \Omega _e \times (0,T) ,\\ u=\partial _t u=0 & \text{ in }{\mathbb R}^n \times \{0\}. \end {cases} \end{aligned} \) The fractional wave equation can be regarded as a special case of the peridynamics which models the nonlocal elasticity theory. Note that the initial boundary value problem (6.1) is a mixed local-nonlocal type equation. We study uniqueness and (optimal) stability results for q in the initial exterior value problem (6.1).