The propagation of magnetic domain walls induced by spin-polarized currents(1-5) has launched new concepts for memory and logic devices(6-8). A wave of studies focusing on permalloy (NiFe) nanowires(9) has found evidence for high domain-wall velocities (100ms(-1); refs 10,11), but has also exposed the drawbacks of this phenomenon for applications. Often the domain-wall displacements are not reproducible(12), their depinning from a thermally stable position is difficult(13) and the domain-wall structural instability (Walker breakdown(14,15)) limits the maximum velocity(10). 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(16) facilitates the depinning and leads to high mobility, while the SIA-mediated Rashba field(17-19) 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).