Graphullerene is a recently discovered, two-dimensional allotrope of carbon formed from C₆₀ molecules. It has been synthesized in the form (C₆₀Mg₄)_n and subsequently transformed into (C₆₀)_n by removal of the Mg atoms. Ab initio calculations are employed to examine the structure and properties of this material. Structurally, graphullerene is composed of strained C₆₀ molecules. Each of these molecules is connected to six neighbors in a hexagonal network with a total of eight chemical bonds. We find this structure to be meta-stable, owing to the strain produced by the covalent bonding of the molecules. However, the inclusion of Mg atoms transforms the cohesion energy from negative to positive values by forming additional C-Mg bonds, creating an energetically stable material. In the absence of Mg, this allotrope is a pure carbon semiconductor with an indirect band gap. Phonon spectrum calculations confirm the dynamical stability of the material and yield its in- and out-of-plane sound velocities. In addition, symmetry breaking of the C₆₀ molecules results in a distribution of bond lengths and creates vibrational modes that serve as a signature of graphullerene.