For the purpose of recovering the intriguing electronic properties of freestanding graphene at a solid surface, graphene self-organized on a Au monolayer on Ni(111) is prepared and characterized by scanning tunneling microscopy. Angle-resolved photoemission reveals a gapless linear $\pi$-band dispersion near $\overline{K}$ as a fingerprint of strictly monolayer graphene and a Dirac crossing energy equal to the Fermi energy ($E_F$) within 25 meV meaning charge neutrality. Spin resolution shows a Rashba effect on the $\pi$ states with a large ($\sim 13\mathrm{meV}$) spin-orbit splitting up to $E_F$ which is independent of $\mathbf{k}$.