Let \(\Omega\subset {\mathbb R}^N\) be a bounded, open set of class \(C^\infty\) and let \(\beta\in C^\infty(\partial\Omega)\) be a non-negative function. The author considers the Laplacian \(\Delta_R\) on \(\Omega\) subject to Robin boundary conditions \({\partial u \over \partial\nu}+\beta u=0\) on \(\partial\Omega\). This operator generates an analytic contractive \(C_0\)-semigroup \((T_R(t))_{t\geq 0}\) on the space \(C(\overline{\Omega})\). Let \(G_0(t)\) denote the \(C_0\)-semigroup on \(C_0({\mathbb R}^N)\) with generator \({\mathcal D}(\Delta_0)=\{u\in C_0({\mathbb R}^N)\mid\Delta u\in C_0({\mathbb R}^N)\}\), \(\Delta_0 u=\Delta u\).
A positive, contractive, linear extension operator \(E_\beta:C(\overline{\Omega})\to C_0({\mathbb R}^N)\) which maps an operator core for \(\Delta_R\) into \({\mathcal D}(\Delta_0)\) is constructed. Let \(R : C_0({\mathbb R}^N)\to C(\overline{\Omega})\) denote the restriction operator \(Ru = u|_{\overline{\Omega}}\). The main result of the paper is the following Trotter product formula: \[ T_R(t)=\lim_{n\to\infty} \left(RG_0\left({t\over n}\right)E_\beta \right)^n. \]