  Neutron scattering is used to study spin waves in the three-dimensional
Heisenberg ferromagnet YTiO$_3$, with spin-spin exchange disorder introduced
$via$ La-substitution at the Y site. No significant changes are observed in the
spin-wave dispersion up to a La concentration of 20%. However, a strong
broadening of the spectrum is found, indicative of shortened spin-wave
lifetimes. Density-functional theory calculations predict minimal changes in
exchange constants as a result of average structural changes due to La
substitution, in agreement with the data. The absence of significant changes in
the spin-wave dispersion, the considerable lifetime effect, and the reduced
ordered magnetic moment previously observed in the La-substituted system are
qualitatively captured by an isotropic, nearest-neighbor, three-dimensional
Heisenberg ferromagnet model with random antiferromagnetic exchange. We
therefore establish Y$_{1-x}$La$_x$TiO$_3$ as a model system to study the
effect of antiferromagnetic spin-exchange disorder in a three-dimensional
Heisenberg ferromagnet.
