Graphene is one of the most promising materials for nanoelectronics owing to its unique Dirac cone-like dispersion of the electronic state and high mobility of the charge carriers. However, to facilitate the implementation of the graphene-based devices, an essential change of its electronic structure, a creation of the band gap should controllably be done. Brought about by two fundamentally different mechanisms, a sublattice symmetry breaking or an induced strong spin-orbit interaction, the band gap appearance can drive graphene into a narrow-gap semiconductor or a 2D topological insulator phase, respectively, with both cases being technologically relevant. The later case, characterized by a spin-orbit gap between the valence and conduction bands, can give rise to the spin-polarized topologically protected edge states. Here, we study the effect of the spin-orbit interaction enhancement in graphene placed in contact with a lead monolayer. By means of angle-resolved photoemission spectroscopy, we show that intercalation of the Pb interlayer between the graphene sheet and the Pt(111) surface leads to formation of a gap of 200 meV at the Dirac point of graphene. Spin-resolved measurements confirm the splitting to be of a spin-orbit nature, and the measured near-gap spin structure resembles that of the quantum spin Hall state in graphene, proposed by Kane and Mele [ Phys. Rev. Lett. 2005, 95, 226801 ]. With a bandstructure tuned in this way, graphene acquires a functionality going beyond its intrinsic properties and becomes more attractive for possible spintronic applications.