  We report measurements of the nuclear spin-lattice and spin-spin relaxation
times of very dilute 3He in solid 4He in the temperature range 0.01 \leq T \leq
0.5 K for densities where anomalies have been observed in torsional oscillator
and shear modulus measurements. We compare the results with the values of the
relaxation times reported by other observers for higher concentrations and the
theory of Landesman that takes into account the elastic properties of the 4He
lattice. A sharp increase in the magnitude of the nuclear spin-lattice
relaxation times compared to the the classical Landesman theory is observed
close to the temperatures where the torsional and shear modulus anomalies are
observed. The NMR results suggest that the tunneling of 3He impurities in the
atomic-scale elastic distortion is affected by the same processes that give
rise to the macroscopic elastic dissipation anomalies.
