  Cosmic string solutions of the abelian Higgs model with conformal coupling to
gravity are shown to exist. The main characteristics of the solutions are
presented and the differences with respect to the minimally coupled case are
studied. An important difference is the absence of Bogomolnyi cosmic string
solutions for conformal coupling. Several new features of the abelian Higgs
cosmic strings of both types are discussed. The most interesting is perhaps a
relation between the angular deficit and the central magnetic field which is
bounded by a critical value.
