As fabrication technology pushes the dimensions of ferromagnetic structures into the nanoscale, understanding the magnetization processes of these structures is of fundamental interest, and key to future applications in hard disk drives, magnetic random access memory and other 'spintronic' devices. Measurements on elongated magnetic nanostructures highlighted the importance of nucleation and propagation of a magnetic boundary, or domain wall, between opposing magnetic domains in the magnetization reversal process. Domain-wall propagation in confined structures is of basic interest and critical to the performance of a recently demonstrated magnetic logic scheme for spintronics. A previous study of a 500-nm-wide NiFe structure obtained very low domain-wall mobility in a three-layer device. Here we report room-temperature measurements of the propagation velocity of a domain wall in a single-layer planar Ni80Fe20 ferromagnetic nanowire 200 nm wide. The wall velocities are extremely high and, importantly, the intrinsic wall mobility is close to that in continuous films, indicating that lateral confinement does not significantly affect the gyromagnetic spin damping parameter to the extreme extent previously suggested. Consequently the prospects for high-speed domain-wall motion in future nanoscale spintronic devices are excellent.