The growth and melting of He crystals spin-polarized by a magnetic field at temperatures below the antiferromagnetic transition (and below the superfluid transition in the liquid) are accompanied by superfluid spin currents in addition to mass flows. In fields weak in comparison with exchange fields, the crystallization waves change in nature, because the spin currents begin to play a predominant role. The wave spectrum acquires an acoustic nature, with a velocity inversely proportional to the magnetic field.