  In zero magnetic field, conductance measurements of clean one-dimensional
(1D) constrictions defined in GaAs/AlGaAs heterostructures show twenty-six
quantized ballistic plateaux, as well as a structure close to $0.7(2e^2/h)$. In
an in-plane magnetic field all the 1D subbands show Zeeman splitting and in the
wide channel limit the $g$-factor is $\mid g \mid = 0.4$, close to that of bulk
GaAs. For the last subband spin-splitting originates from the structure at
$0.7(2e^2/h)$, indicating spin polarization at $B=0$. The measured enhancement
of the $g$-factor as the subbands are depopulated suggests that the ``0.7
structure'' is induced by electron-electron interactions.
