Let \(X\) be a Fréchet space and \(\mu\) a Radon probability measure on \(X\). \(a\in X\) is called the barycenter of \(\mu\) if \(\Lambda a=\int_X\Lambda x\mu(dx)\) for any \(\Lambda\in X^*\). Let \(M\) be a non-empty compact convex subset of \(X\). The authors characterize the set of points \(a\in M\) such that \(a\) is the barycenter of some Radon probability measure \(\mu\) on \(X\) with \(\text{supp }\mu=M\). In particular, if \(X\) is finite-dimensional, this set of barycenters coincides with the relative algebraic interior of \(M\). A counterexample shows that this is not true in infinite-dimensional Hilbert spaces. Moreover, the authors describe the set of barycenters of measures in the case that \(X=\mathcal{M}(K)\) is the space of signed finite Radon measures on a metric compact space \(K\).