In this paper we generalise the concept of a Steinbergcross section to non-connected affine Kac--Moody groups.This Steinberg cross section is a section to therestriction of the adjoint quotient map to a given exteriorconnected component of the affine Kac--Moody group.(The adjoint quotient is only defined on a certain submonoid of theentire group, however, the intersection of this submonoid with eachconnected component is non-void.)The image of the Steinberg cross section consists of a``twisted Coxeter cell'',a transversal slice to a twisted Coxeter element.A crucial point in the proof of the main result is that the image ofthe cross section can be endowed with a $Cst$-action.
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