In this paper, we study the stability of an elastic string system with local Kelvin-Voigt damping. Few results for this system are known in the literature. Under the assumption that the damping coefficient has a singularity at the interface of the damped and undamped regions and behaves like x(alpha) near the interface, we prove that the semigroup corresponding to the system is polynomially or exponentially stable and the decay rate depends on the parameter alpha is an element of (0, 1].