  Properties of the orbitally excited (L=1) $B_s$ states are studied using 1.0
fb$^{-1}$ of pp collisions at $\sqrt{s} =7$ 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 $B_{s1}(5830)^0 \to B^{*+} K^-$ 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 $B_{s2}^*(5840)^0 \to B^{*+} K^-$
decay favours the spin-parity assignment $J^P = 2^+$ for the $B_{s2}^*(5840)^0$
meson. In addition, the most precise measurement of the mass difference
$m(B^{*+})-m(B^+) = 45.01 \pm 0.30 (stat) \pm 0.23 (syst)$ MeV/$c^2$ is
obtained.
