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The stability of the ordered phase of the three-dimensional XY-model withrandom phase shifts is studied by considering the roughening of a singlestretched vortex line due to the disorder. It is shown that the vortex line maybe described by a directed polymer Hamiltonian with an effective randompotential that is long range correlated. A Flory argument estimates theroughness exponent to $\zeta=3/4$ and the energy fluctuation exponent to$\omega=1/2$, thus fulfilling the scaling relation $\omega=2\zeta-1$. TheSchwartz-Edwards method as well as a numerical integration of the correspondingBurger's equation confirm this result. Since $\zeta<1$ the ordered phase of theoriginal XY-model is stable.Comment: 8 pages RevTeX, 3 eps-figures include

Vortex lines in the three-dimensionalXYmodel with random phase shifts