C denotes either the conformal group in 3+1 dimensions, or in one chiral dimension. Let U be a unitary, strongly continuous representation of C satisfying the spectrum condition and inducing, by its adjoint action, automorphisms of a v.Neumann algebra A. We construct the unique inner representation $U^A$ of the universal covering group of C implementing these automorphisms. $U^A$ satisfies the spectrum condition and acts trivially on any U-invariant vector. This means in particular: Conformal transformations of a field theory having positive energy are weak limit points of local observables. Some immediate implications for chiral subnets are given. We propose the name ``Borchers-Sugawara construction''.