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Dirac cones in a two-dimensional environment have attracted much attention not only because of the massless Dirac fermions but also due to their capability to lock the spin direction with the momentum. Here we demonstrate that the Rashba effect within a single layer of a binary alloy composed of heavy atoms, Pb and Au, can be driven by and even tweaked with the adjacent top and bottom layers to yield cone-like structures and further enhance the Rashba coupling strength. Two cones are observed at the surface zone center $\bar{\Gamma }$ with giant Rashba parameters 1.53 and 4.45 eVÅ; an anisotropic giant Rashba splitting at the surface zone boundary $\bar{M}$ has a great value, 6.26 eVÅ, inferring the critical role of p - d hybridization between Pb and Au. Our results reveal not only an interesting natural phenomenon but also a feasible method of tweaking the Rashba effect of a two-dimensional system.