  A self-consistent general relativistic configuration describing a finite
cross-section magnetic flux tube is constructed. The cosmic solenoid is modeled
by an elastic superconductive surface which separates the Melvin core from the
surrounding flat conic structure. We show that a given amount $\Phi$ of
magnetic flux cannot be confined within a cosmic solenoid of circumferential
radius smaller than $\frac{\sqrt{3G}}{2\pi c^2}\Phi$ without creating a conic
singularity. Gauss-Codazzi matching conditions are derived by means of a
self-consistent action. The source term, representing the surface currents, is
sandwiched between internal and external gravitational surface terms. Surface
superconductivity is realized by means of a Higgs scalar minimally coupled to
projective electromagnetism. Trading the 'magnetic' London phase for a dual
'electric' surface vector potential, the generalized quantization condition
reads: $e/{hc} \Phi + 1/e Q=n$ with $Q$ denoting some dual 'electric' charge,
thereby allowing for a non-trivial Aharonov-Bohm effect. Our conclusions
persist for dilaton gravity provided the dilaton coupling is sub-critical.
