Theoretical calculations, based on the hybrid exchange density functional theory, are used to show that in graphene, a periodic array of defects generates a ferromagnetic ground state at room temperature for unexpectedly large defect separations. This is demonstrated for defects that consist of a carbon vacancy in which two of the dangling bonds are saturated with H atoms. The magnetic coupling mechanism is analysed and found to be due to an instability in the π-electron system with respect to a long-range spin polarization characterized by alternation in the spin direction between adjacent carbon atoms. The disruption of the π-bonding opens a semiconducting gap at the Fermi edge. The size of the energy gap and the magnetic coupling strength are strong functions of the defect separation and can thus be controlled by varying the defect concentration. The position of the semiconducting energy gap and the electron effective mass are strongly spin-dependent and this is expected to result in a spin asymmetry in the transport properties of the system. A defective graphene sheet is, therefore, a very promising material with an in-built mechanism for tailoring the properties of future spintronic devices.