The propagation of magnetic domain walls induced by spin-polarized currents has launched new concepts for memory and logic devices. A wave of studies focusing on permalloy (NiFe) nanowires has found evidence for high domain-wall velocities (100 m s(-1); refs,), but has also exposed the drawbacks of this phenomenon for applications. Often the domain-wall displacements are not reproducible, their depinning from a thermally stable position is difficult and the domain-wall structural instability (Walker breakdown) limits the maximum velocity. Here, we show that the combined action of spin-transfer and spin-orbit torques offers a comprehensive solution to these problems. In an ultrathin Co nanowire, integrated in a trilayer with structural inversion asymmetry (SIA), the high spin-torque efficiency facilitates the depinning and leads to high mobility, while the SIA-mediated Rashba field controlling the domain-wall chirality stabilizes the Bloch domain-wall structure. Thus, the high-mobility regime is extended to higher current densities, allowing domain-wall velocities up to 400 m s(-1).