The properties of the orbitally excited (L=1) Bs(0) states are studied by using 1.0 fb(-1) of pp collisions at sqrt[s]=7 TeV collected with the LHCb detector. The first observation of the Bs2*(5840)(0) meson decaying to B*+ K- is reported, and the corresponding branching fraction measured relative to the B+ K- decay mode. The Bs1(5830)(0)→B*+ K- decay is observed as well. The width of the Bs2*(5840)(0) state is measured for the first time, and the masses of the two states are determined with the highest precision to date. The observation of the Bs2*(5840)(0)→B*+ K- decay favors the spin-parity assignment J(P)=2+ for the Bs2*(5840)(0) meson. In addition, the most precise measurement of the mass difference m(B*+)-m(B+)=45.01±0.30(stat)±0.23(syst) MeV/c(2) is obtained.