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.