  If the graviton possesses an arbitrarily small (but nonvanishing) mass,
perturbation theory implies that cosmic strings have a nonzero Newtonian
potential. Nevertheless in Einstein gravity, where the graviton is strictly
massless, the Newtonian potential of a cosmic string vanishes. This discrepancy
is an example of the van Dam--Veltman--Zakharov (VDVZ) discontinuity. We
present a solution for the metric around a cosmic string in a braneworld theory
with a graviton metastable on the brane. This theory possesses those features
that yield a VDVZ discontinuity in massive gravity, but nevertheless is
generally covariant and classically self-consistent. Although the cosmic string
in this theory supports a nontrivial Newtonian potential far from the source,
one can recover the Einstein solution in a region near the cosmic string. That
latter region grows as the graviton's effective linewidth vanishes (analogous
to a vanishing graviton mass), suggesting the lack of a VDVZ discontinuity in
this theory. Moreover, the presence of scale dependent structure in the metric
may have consequences for the search for cosmic strings through gravitational
lensing techniques.
