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Interfacial interactions allow the electronic properties of graphene to be modified, as recently demonstrated by the appearance of satellite Dirac cones in graphene on hexagonal boron nitride substrates. Ongoing research strives to explore interfacial interactions with other materials to engineer targeted electronic properties. Here we show that with a tungsten disulfide (WS2) substrate, the strength of the spin-orbit interaction (SOI) in graphene is very strongly enhanced. The induced SOI leads to a pronounced low-temperature weak anti-localization effect and to a spin-relaxation time two to three orders of magnitude smaller than in graphene on conventional substrates. To interpret our findings we have performed first-principle electronic structure calculations, which confirm that carriers in graphene on WS2 experience a strong SOI and allow us to extract a spin-dependent low-energy effective Hamiltonian. Our analysis shows that the use of WS2 substrates opens a possible new route to access topological states of matter in graphene-based systems.