  We prove that in genus bigger than $2$, the mapping class group action on
$\mathrm{Aff}(\mathbb{C})$-characters is ergodic. This implies that almost
every representation $\pi_1 S \longrightarrow \mathrm{Aff}(\mathbb{C})$ is the
holonomy of a branched affine structure on $S$, where $S$ is a closed
orientable surface of genus $g \geq 2$.
