We produce affine interval exchange transformations (AIETs) which are topologically conjugated to (standard) interval exchange maps (IETs) via a singular conjugacy, i.e. a diffeomorphism h of [0,1] which is C^0 but not C^1 and such that the pull-back of the Lebesgue measure is a singular invariant measure for the AIET. In particular, we show that for almost every IET T_0 of at least two intervals and any vector w belonging to the central-stable space E_{cs}(T_0) for the Rauzy-Veech renormalization, any AIET T with log-slopes given by w and semi-conjugated to T_0 is topologically conjugated to T. If in addition, if w does not belong to E_s(T_0), the conjugacy between T and T_0 is singular.
