  Non-equilibrium excess noise of a short quasi one-dimensional constriction
between two superconductors is considered. A general expression for the
current-current correlation function valid for arbitrary temperatures and bias
voltages is derived. This formalism is applied to a current-carrying quantum
channel with perfect transparency. Contrary to a transparent channel separating
two normal conductors, a weak link between two superconductors exhibits a
finite level of noise. The source of noise is fractional Andreev scattering of
quasiparticles with energies $|E|$ greater than the half-width $\Delta$ of the
superconducting gap. For high bias voltages, $V \gg \Delta /e$, the relation
between the zero-frequency limit of the noise spectrum, $S(0)$, and the excess
current $I_{\text{exc}}$ reads $S(0)=(1/5)|e|I_{\text{exc}}$. As $\Delta
\rightarrow 0$ both the excess noise and the excess current vanish linearly in
$\Delta$, %$\propto \Delta$, their ratio being constant.
