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A theory is presented to describe quantitatively how nuclear spin polarization can be induced in a metal by a dc electric bias current driven through the sample. The bias current is spin polarized by passing through a thin ferromagnetic film in interfacial contact with the sample, with the polarization axis determined by the magnetization orientation of the ferromagnetic layer. The nuclear spins are polarized due to spin angular momentum transfer from the electrons to the nuclei mediated by the contact hyperfine interaction. Upon reversing the polarity of the bias current, the sign of the nuclear polarization is reversed. Unlike other methods of dynamic nuclear polarization, spin-injected dynamic nuclear polarization is a zero frequency technique that in principle does not require an applied magnetic field.