  We study spatial variations of the fine-structure constant in the presence of
static straight cosmic strings in the weak-field approximation in Einstein
gravity. We work in the context of a generic Bekenstein-type model and consider
a gauge kinetic function linear in the scalar field. We determine an analytical
form for the scalar field and the string metric at large distances from the
core. We show that the gravitational effects of $\alpha$-varying strings can be
seen as a combination of the gravitational effects of global and local strings.
We also verify that at large distances to the core the space-time metric is
similar to that of a global string. We study the motion of test particles
approaching from infinity and show that photons are scattered to infinity while
massive particles are trapped in bounded trajectories. We also calculate an
overall limit on the magnitude of the variation of $\alpha$ for a GUT string,
by considering suitable cosmological constraints coming from the Equivalence
Principle.
