  Let $Y$ be the underlying variety of a connected affine algebraic group. We
prove that two embeddings of the affine line $\mathbb{C}$ into $Y$ are the same
up to an automorphism of $Y$ provided that $Y$ is not isomorphic to a product
of a torus $(\mathbb{C}^\ast)^k$ and one of the three varieties $\mathbb{C}^3$,
$\operatorname{SL}_2$, and $\operatorname{PSL}_2$.
