  Let $\mathcal{D}$ be a non-trivial $G$-block-transitive $3$-$(v,k,1)$ design,
where $T\leq G \leq \mathrm{Aut}(T)$ for some finite non-abelian simple group
$T$. It is proved that if $T$ is a simple exceptional group of Lie type, then
$T$ is either the Suzuki group ${}^2B_2(q)$ or $G_2(q)$. Furthermore, if
$T={}^2B_2(q)$ then the design $\mathcal{D}$ has parameters $v=q^2+1$ and
$k=q+1$, and so $\mathcal{D}$ is an inverse plane of order $q$; and if
$T=G_2(q)$ then the point stabilizer in $T$ is either $\mathrm{SL}_3(q).2$ or
$\mathrm{SU}_3(q).2$, and the parameter $k$ satisfies very restricted
conditions.
