  Recently, the chiral superconductivity of the cosmic string in the axion
model has gathered attention. The superconductive nature can alter the standard
understanding of the cosmology of the axion model. For example, a string loop
with a sizable superconducting current can become a stable configuration, which
is called a Vorton. The superconductive nature can also affect the cosmological
evolution of the string network. In this paper, we study the stability of the
superconducting current in the string. We find the superconductivity is indeed
stable for a straight string or infinitely small string core size, even if the
carrier particles are unstable in the vacuum. However we also find that the
carrier particle decays in a curved string in typical axion models, if the
carrier particles are unstable in the vacuum. Accordingly, the lifetime of the
Vorton is not far from that of the carrier particle in the vacuum.
