We report a novel crystalline carbon-cage structure synthesized from laser-driven shock wave loading of a graphite-copper mixture to about 14+/-2 GPa and 1000 +/- 200 K. Quite unexpectedly, it can be structurally related to an extremely compressed three-dimensional C60 polymer with random displacement of C atoms around average positions equivalent to those of distorted C60 cages. Thus, the present carbon-cage structure represents a structural crossing point between graphite interlayer bridging and C60 polymerization as the two ways of forming diamond from two-dimensional and molecular carbon.