The properties of the orbitally excited (L=1) B0s states are studied by using 1.0 fb−1 of pp collisions at s√=7 TeV collected with the LHCb detector. The first observation of the B∗s2(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 B∗s2(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 B∗s2(5840)0→B*+K− decay favors the spin-parity assignment JP=2+ for the B∗s2(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/c2 is obtained.