Summary: We prove the uniqueness for the inverse problem of determining a coefficient \(q(x)\) in \[ \partial^2_tu(x,t)=\Delta u(x,t)-q(x)u(x,t) \] for \(x\in \mathbb{R}^n\) and \(t>0\), from observations of \(u |_{\Gamma\times(0,T)}\) and the normal derivative \(\frac{\partial u} {\partial v}|_{\Gamma\times(0,T)}\) where \(\Gamma\) is an arbitrary \(C^\infty\)-hypersurface. Our main result asserts the uniqueness of \(q\) over \(\mathbb{R}^n\) provided that \(T>0\) is sufficiently large and \(q\) is analytic near \(\Gamma\) and outside a ball. The proof depends on Fritz John’s global Holmgren theorem and the uniqueness by a Carleman estimate.