The mine in question is the Mir mine in the Mirniy industrial zone in Eastern Siberia. You might have probably seen photos of it before, labelled "the largest man-made hole on earth" or something similar. It's long been used to mine diamonds, but the mining stopped in June 2001, replaced with a network of underground tunnels below. Presumably this activity would have to cease if the structure is ever built.
The project, which is described in a series of slides on AB Elise's site, is split between three levels, penetrated in the centre by a vertical farm and forests. Around the exterior are residences and recreational areas, and there's also a research centre in the plans.