We observe the decays D*_(s1)(2710)^+ → D*K and D*_(sJ)(2860)^+ → D*K and measure their branching fractions relative to the DK final state. We also observe, in the D*K mass spectrum, a new broad structure at a mass of (3044 ± 8_(stat)(^(+30)_(-5))_(syst)) MeV/c^2 having a width Γ =(239 ± 35_(stat)(^(+46)_(-42))_(syst)) MeV. To obtain this result we use 470 fb^(-1) of data recorded by the BABAR detector at the PEP-II asymmetric-energy e^+e^- storage rings at the Stanford Linear Accelerator Center running at center-of-mass energies near 10.6 GeV
