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This paper proves the Corona Theorem to be affirmative for domains in the complex plane bounded by thick subsets of a Lipschitz graph. Specifically, the boundary of these domains E 0 has a Carleson lower density: Λ (B(z, r) ∩ E 0 ) > ǫ 0 r for all z ∈ E 0 , and all r > 0. 2000 Mathematics Subject Classification. 30-XX.doi:10.5565/publmat_55111_05 fatcat:2z3em7vdmzci3geemhbij6dzhe