  Let $\mathcal{D}$ be a nontrivial $3$-$(v,k,1)$ design admitting a
block-transitive group $G$ of automorphisms. A recent work of Gan and the
second author asserts that $G$ is either affine or almost simple. In this
paper, it is proved that if $G$ is almost simple with socle an alternating
group, then $\mathcal{D}$ is the unique $3$-$(10,4,1)$ design, and
$G=\mathrm{PGL}(2,9)$, $\mathrm{M}_{10}$ or $\mathrm{Aut}(\mathrm{A}_6
)=\mathrm{S}_6:\mathrm{Z}_2$, and $G$ is flag-transitive.
