 Amenability of a free product

For nontrivial groups $A,B$, the free product $A\ast B$ is amenable exactly in the exceptional case $A\cong B\cong C_2$. In that case it is the infinite dihedral group, hence a virtually cyclic group and amenable. In every other case the normal form theorem for a free product and the ping-pong lemma produce a subgroup isomorphic to $F_2$; the fact that a subgroup of an amenable group is amenable then obstructs amenability.