The Ray-Singer theorem [D. B. Ray, I. M. Singer, Ann. Math. (2) 98, 154–177 (1973; Zbl 0267.32014)] factorizes the determinant of the Laplacian corresponding to the smooth flat metric \(\arrowvert w \arrowvert ^{2}\), obtained from the Abelian differential \(w\), on an elliptic curve (that is, a Riemann surface of genus one). This article generalizes the Ray-Singer result by giving a factorization of this determinant on Riemann surfaces of genus \(g>1\), involving Bergman tau-functions in place of the Dedekind eta-function. This is achieved via extensive and explicit computation of variational formulae on spaces of Abelian differentials over Riemann surfaces and determinants of Laplacians on surfaces with flat conical metrics. Further generalizations of the Ray-Singer formula are referenced at the end of the article.