  An angle-resolved x-ray spectroscopic scheme is presented for determining the
hyperfine splitting of highly charged ions. For helium-like ions, in
particular, we propose to measure either the angular distribution or
polarization of the $1s2p~^{3}P_{1}, F \rightarrow 1s^{2}~^{1}S_{0}, F_{f}$
emission following the stimulated decay of the initial $1s2s~^{1}S_{0}, F_{i}$
level. It is found that both the angular and polarization characteristics of
the emitted x-ray photons strongly depends on the (relative) \textit{splitting}
of the partially overlapping hyperfine $1s2p~^{3}P_{1}, F$ resonances and may
thus help resolve their hyperfine structure. The proposed scheme is feasible
with present-day photon detectors and allows a measurement of the hyperfine
splitting of helium-like ions with a relative accuracy of about $10^{-4}$.
