  We predict that spin-waves in an ordered quantum antiferromagnet (AFM) in a
strong magnetic field become unstable with respect to spontaneous two-magnon
decays. At zero temperature, the instability occurs between the threshold field
$H^*$ and the saturation field $H_c$. As an example, we investigate the
high-field dynamics of a Heisenberg antiferromagnet on a square lattice and
show that the single-magnon branch of the spectrum disappears in the most part
of the Brillouin zone.
