  Nuclear magnetic resonance (NMR) and transport measurements have been
performed at high magnetic fields and low temperatures in a series of $n$-type
Bi$_{2}$Se$_{3}$ crystals. In low density samples, a complete spin polarization
of the electronic system is achieved, as observed from the saturation of the
isotropic component of the $^{209}$Bi NMR shift above a certain magnetic field.
The corresponding spin splitting, defined in the phenomenological approach of a
3D electron gas with a large (spin-orbit-induced) effective $g$-factor, scales
as expected with the Fermi energy independently determined by simultaneous
transport measurements. Both the effective electronic $g$-factor and the
"contact" hyperfine coupling constant are precisely determined. The magnitude
of this latter reveals a non negligible $s$-character of the electronic wave
function at the bottom of the conduction band. Our results show that the bulk
electronic spin polarization can be directly probed via NMR and pave the way
for future NMR investigations of the electronic states in Bi-based topological
insulators.
