We consider the unbounded settling dynamics of a circular disk of diameter d and finite thickness h evolving with a vertical speed U in a linearly stratified fluid of kinematic viscosity v and diffusivity k of the stratifying agent, at moderate Reynolds numbers (Re=Ud/v ). The influence of the disk geometry (diameter d and aspect ratio x = d/h ) and of the stratified environment (buoyancy frequency N, viscosity and diffusivity) are experimentally and numerically investigated. Three regimes for the settling dynamics have been identified for a disk reaching its gravitational equilibrium level. The disk first falls broadside-on, experiencing an enhanced drag force that can be linked to the stratification. A second regime corresponds to a change of stability for the disk orientation, from broadside-on to edgewise settling. This occurs when the non-dimensional velocity U/√vN becomes smaller than some threshold value. Uncertainties in identifying the threshold value is discussed in terms of disk quality. It differs from the same problem in a homogeneous fluid which is associated with a fixed orientation (at its initial value) in the Stokes regime and a broadside-on settling orientation at low, but finite Reynolds numbers. Finally, the third regime corresponds to the disk returning to its broadside orientation after stopping at its neutrally buoyant level.