  This paper determined all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a
non-symmetric 2-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G$ is the
almost simple flag-transitive automorphism group of $\mathcal{D}$ with an
exceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$
where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group
or Suzuki group, and there are five classes of non-isomorphic designs
$\mathcal{D}$.
