  We study the modifications on the metric of an isolated self-gravitating
bosonic superconducting cosmic string in a scalar-tensor gravity in the
weak-field approximation. These modifications are induced by an arbitrary
coupling of a massless scalar field to the usual tensorial field in the
gravitational Lagrangian. The metric is derived by means of a matching at the
string radius with a most general static and cylindrically symmetric solution
of the Einstein-Maxwell-scalar field equations. We show that this metric
depends on five parameters which are related to the string's internal structure
and to the solution of the scalar field. We compare our results with those
obtained in the framework of General Relativity.
