Summary: We establish the best constant for a Sobolev trace inequality on compact Riemannian manifolds with boundary. More specifically, let \(1< p< n\) and \[ {1\over\overline K(n,p)}= \inf_{\substack{\nabla u\in L^p(\mathbb{R}^n_+)\\u\in L^{\overline p^*}(\partial\mathbb{R}^n_+)\setminus \{0\}}} {\int_{\mathbb{R}^n_+}|\nabla u|^p\over (\int_{\partial\mathbb{R}^n_+}|u|^{\overline p^*})^{p,\overline p^*}}, \] where \(\overline p^*= p(n- 1)/(n- p)\). We prove that for any compact \(n\)-dimensional Riemannian manifold with boundary \((M,g)\), for any \(\varepsilon> 0\) there exists \(A_\varepsilon> 0\) such that \[ \|u\|^p_{L^{\overline p^*}(\partial M)}\leq (\overline K(n, p)+ \varepsilon)\|\nabla_g u\|^p_{L^p(M)}+ A_\varepsilon\|u\|^p_{L^p(\partial M)} \] for all \(u\in H^{1,p}(M)\).