We show that if $G$ is a countable amenable group, then every stationary non-Gaussian symmetric $α$-stable (S$α$S) process indexed by $G$ is ergodic if and only if it is weakly-mixing, and it is ergodic if and only if its Rosinski minimal spectral representation is null. This extends the results for $\mathbb{Z}^d$, and answers a question of P. Roy on discrete nilpotent groups to the extent of all countable amenable groups. As a result we construct on the Heisenberg group and on many Abelian groups, for all $α$ in (0,2), stationary S$α$S processes that are weakly-mixing but not strongly-mixing.