The authors prove that the spaces \(\mathcal M\) and \({\mathcal M}_{\text{Lip}}'\) of smooth (resp. nondegenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions, have the structure of infinite-dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite-dimensional version of a theorem of Kirwan, Kempf and Ness relating symplectic quotients to holomorphic quotients, applied to the action of \(\text{PSL}_2(\mathbb{C})\) on the free loop space of the 2-sphere. A key role is played by the conformal center of mass of \textit{A. Douady} and \textit{C. J. Earle} [Acta Math. 157, 23-48 (1986; Zbl 0615.30005)], of which notion the authors give a self-contained treatment in the present paper.