There is an ongoing debate concerning the distribution of eddy stirring across the Antarctic Circumpolar Current (ACC) and the nature of its controlling processes. The problem is addressed here by estimating the isentropic eddy diffusivity kappa from a collection of hydrographic and altimetric observations, analyzed in a mixing length theoretical framework. It is shown that, typically, kappa is suppressed by an order of magnitude in the upper kilometer of the ACC frontal jets relative to their surroundings, primarily as a result of a local reduction of the mixing length. This observation is reproduced by a quasi-geostrophic theory of eddy stirring across a broad barotropic jet based on the scaling law derived by Ferrari and Nikurashin (2010). The theory interprets the observed widespread suppression of the mixing length and kappa in the upper layers of frontal jets as the kinematic consequence of eddy propagation relative to the mean flow within jet cores. Deviations from the prevalent regime of mixing suppression in the core of upper-ocean jets are encountered in a few special sites. Such 'leaky jet' segments appear to be associated with sharp stationary meanders of the mean flow that are generated by the interaction of the ACC with major topographic features. It is contended that the characteristic thermohaline structure of the Southern Ocean, consisting of multiple upper-ocean thermohaline fronts separated and underlaid by regions of homogenized properties, is largely a result of the widespread suppression of eddy stirring by parallel jets.