  We predict that spin-waves in an easy-plane ferromagnet have a finite
lifetime at zero temperature due to spontaneous decays. In zero field the
damping is determined by three-magnon decay processes, whereas decays in the
two-particle channel dominate in a transverse magnetic field. Explicit
calculations of the magnon damping are performed in the framework of the
spin-wave theory for the $XXZ$ square-lattice ferromagnet with an anisotropy
parameter $\lambda<1$. In zero magnetic field the decays occur for
$\lambda^*<\lambda<1$ with $\lambda^*\approx 1/7$. We also discuss possibility
of experimental observation of the predicted effect in a number of
ferromagnetic insulators.
