The dominant decay pathways of argon $2{p}^{\ensuremath{-}2}$ double-core-hole states have been investigated using synchrotron radiation and a magnetic-bottle-type spectrometer coupled with an ion time-of-flight spectrometer. This experiment allows for efficient multi-electron-ion coincidence measurements, and thus for following the Auger cascade step by step in detail. Dominant decay pathways leading to ${\mathrm{Ar}}^{4+}$ final states via ${\mathrm{Ar}}^{3+}$ intermediate states have been assigned with the help of theoretical ab initio calculations. The weak correlated decay of the two core holes by emission of a single Auger electron, leading to ${\mathrm{Ar}}^{3+}$ final states, has been observed at 458.5-eV kinetic energy. Compared to the total decay of the $2{p}^{\ensuremath{-}2}$ double core vacancies, this two-electron--one-electron process was measured to have a branching ratio of $1.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\ifmmode\pm\else\textpm\fi{}1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. Furthermore, the remaining decay paths of the ${\mathrm{Ar}}^{1+}\phantom{\rule{0.28em}{0ex}}(1{s}^{\ensuremath{-}1})$ core hole to higher charge states and their respective contributions to the total yield have been analyzed and show very good agreement with theoretical results.