The authors compute a minimal model of a smooth \(S^1\)-manifold \(M\) in terms of the orbit space \(B=M/S^1\) and the fixed point set \(F\subset B\). In case \(F\) is empty, such computation is presented in [P. A. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Progr. Math. 16 (1981; Zbl 0474.55001)] to the effect that a dgc algebra model of \(M\) is given by a Hirsch extension of the dgc algebra Sullivan minimal model \({\mathcal A}(B)\) of \(B\). In case \(F\) is non-empty, this approach does not apply and the authors of the article compute a minimal model of the deRham dgc algebra \(\Omega(M)\) of \(M\), which is a dg module over the Sullivan minimal model \({\mathcal A}(B)\) of \(B\). In case \(F\) is empty, the formula they obtain coincides with the previous one. By considering the \({\mathcal A}(B)\)-dg module structure associated to \(F\) by means of the natural inclusion \(F\subset B\), the authors also compute the minimal model of \(F\) regarded as an \({\mathcal A}(B)\)-dg module. Moreover, they compute the minimal model of the Borel space \(M\times_{S^1}S^\infty\), where an \({\mathcal A}(B)\)-dg module structure is defined by making use of the canonical projection \(M\times_{S^1} S^\infty\to B\).