An atomic carbon chain joining two graphene flakes was recently realized in a ground-breaking experiment by Jin et al. (Phys. Rev. Lett., 102 (2009) 205501). We present ab initio results for the electron transport properties of such chains and demonstrate complete spin-polarization of the transmission in large energy ranges. The effect is due to the spin-polarized zig-zag edge terminating each graphene flake causing a spin-splitting of the graphene pi(z) bands, and the chain states. Transmission occurs when the graphene p-states resonate with similar states in the strongly hybridized edges and chain. This effect should in general hold for any p-conjugated molecules bridging the zig-zag edges of graphene electrodes. The polarization of the transmission can be controlled by chemically or mechanically modifying the molecule, or by applying an electrical gate.