- We study current-induced magnetization dynamics in a long thin ferromagnetic wire with a Dzyaloshinskii-Moriya interaction (DMI). We find a spiral domain wall configuration of the magnetization and obtain an analytical expression for the width of the domain wall as a function of the interaction strengths. Our findings show that above a certain value of DMI a domain wall configuration cannot exist in the wire. Below this value we determine the domain wall dynamics for small currents, and calculate the drift velocity of the domain wall along the wire. We show that the DMI suppresses the minimum value of current required to move the domain wall. Depending on its sign, the DMI increases or decreases the domain wall drift velocity.