We report a Cu NMR study performed under high magnetic field on single crystals of the inorganic spin-Peierls CuGeO 3 compound, in its uniform and dimerized phases. The temperature (T) dependence of the magnetic hyperfine shift, i.e., the local static spin susceptibility, is found to scale the macroscopic susceptibility. This allows the determination of the hyperfine coupling tensor, which can be well accounted for by the on-site coupling associated with the localized electronic spin of a Cu ϩϩ ion in an approximately axially symmetric environment. The nuclear spin-lattice relaxation rate ͑T 1 Ϫ1 ͒ in the dimerized phase is found to be activated for temperature below ϳT SP /2. The magnetic-field dependence of the activation energy is in good agreement with that of the energy gap determined by neutron inelastic scattering. In the uniform phase, neither the magnitude of T 1 Ϫ1 ͑an order of magnitude smaller͒ nor its T dependence ͑approximately linear in T instead of being nearly constant͒ correspond to the theoretical predictions for a simple Sϭ1/2, one-dimensional Heisenberg antiferromagnet, unless we admit for a filtering of the antiferromagnetic fluctuations, due to a small supertransferred hyperfine coupling. As expected, we found no magnetic-field dependence of T 1 ͑Tտ1.5 T SP ͒ in the range 8.9-14.9 T.