Controllable production of nanometre-sized structures is an important field of research, and synthesis of one-dimensional objects, such as nanowires, is a rapidly expanding area with numerous applications, for example, in electronics, photonics, biology and medicine. Nanoscale electronic devices created inside nanowires, such as p-n junctions, were reported ten years ago. More recently, hetero-structure devices with clear quantum-mechanical behaviour have been reported, for example the double-barrier resonant tunnelling diode and the single-electron transistor. The generally accepted theory of semiconductor nanowire growth is the vapour-liquid-solid (VLS) growth mechanism, based on growth from a liquid metal seed particle. In this letter we suggest the existence of a growth regime quite different from VLS. We show that this new growth regime is based on a solid-phase diffusion mechanism of a single component through a gold seed particle, as shown by in situ heating experiments of GaAs nanowires in a transmission electron microscope, and supported by highly resolved chemical analysis and finite element calculations of the mass transport and composition profiles.