We study the global U(1) string coupled to gravity in the presence of
a negative cosmological constant. We obtain the static straight-string
solution with no curvature singularity. The (2 + 1)D spacetime orthog
onal to the string direction is characterized by the regular hyperbolo
id, the extremal black hole with one horizon or the charged black hole
with two horizons according to the scale of the cosmological constant
. The relationship between the topological charge of a neutral global
string and the black-hole charge is clarified by a duality transformat
ion. The cosmological relevance of these solutions is briefly discusse
d.