  We characterize the contribution from accreted material to the galactic discs
of the Auriga Project, a set of high resolution magnetohydrodynamic
cosmological simulations of late-type galaxies performed with the moving-mesh
code AREPO. Our goal is to explore whether a significant accreted (or ex-situ)
stellar component in the Milky Way disc could be hidden within the
near-circular orbit population, which is strongly dominated by stars born
in-situ. One third of our models shows a significant ex-situ disc but this
fraction would be larger if constraints on orbital circularity were relaxed.
Most of the ex-situ material ($\gtrsim 50\%$) comes from single massive
satellites ($> 6 \times 10^{10}~M_{\odot}$). These satellites are accreted with
a wide range of infall times and inclination angles (up to $85^{\circ}$).
Ex-situ discs are thicker, older and more metal-poor than their in-situ
counterparts. They show a flat median age profile, which differs from the
negative gradient observed in the in-situ component. As a result, the
likelihood of identifying an ex-situ disc in samples of old stars on
near-circular orbits increases towards the outskirts of the disc. We show three
examples that, in addition to ex-situ discs, have a strongly rotating dark
matter component. Interestingly, two of these ex-situ stellar discs show an
orbital circularity distribution that is consistent with that of the in-situ
disc. Thus, they would not be detected in typical kinematic studies.
