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Transverse spin dynamics have been measured for a very dilute $^{3}\mathrm{\ensuremath{-}}^{4}$He mixture (${\mathit{x}}_{3}$=1.82\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$) polarized by an 8-T magnetic field. The sample was confined to a small spherical resonator, resulting in well-defined spin-wave modes for temperatures below 50 mK. Results are presented for the spin-rotation parameter ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{int}}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$ and spin-wave dispersion ${\mathit{D}}_{\mathrm{\ensuremath{\perp}}}$/${\mathrm{\ensuremath{\Omega}}}_{\mathrm{int}}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$. It is found that ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{int}}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$ increases monotonically as the temperature is decreased below the degeneracy temperature ${\mathit{T}}_{\mathit{F}}$, in apparent disagreement with an earlier experiment.