We show that two graphene layers stacked directly on top of each other (AA stacking) form strong chemical bonds when the distance between planes is 0.156 nm. Simultaneously, C-C in-plane bonds are considerably weakened from partial double-bond (0.141 nm) to single bond (0.154 nm). This polymorphic form of graphene bilayer is meta-stable w.r.t. the one bound by van der Waals forces at a larger separation (0.335 nm) with an activation energy of 0.16 eV/cell. Similarly to the structure found in hexaprismane, C forms four single bonds in a geometry mixing 90 and 120 angles. Intermediate separations between layers can be stabilized under external anisotropic stresses showing a rich electronic structure changing from semi-metal at van der Waals distance, to metal when compressed, to wide gap semiconductor at the meta-stable minimum.Financed by CICYT under contracts MAT-2005-3866, MAT-2006-03741, FIS-2006-12117-C04-03,NAN-2004-09183-C10-08. We acknowledge the use of the Spanish Supercomputing Network and the CTI (CSIC).Peer reviewed