Kimberlite magmas have the deepest origin of all terrestrial magmas and are exclusively associated with cratons(1-3). During ascent, they travel through about 150 kilometres of cratonic mantle lithosphere and entrain seemingly prohibitive loads (more than 25 per cent by volume) of mantle-derived xenoliths and xenocrysts (including diamond)(4,5). Kimberlite magmas also reputedly have higher ascent rates(6-9) than other xenolith-bearing magmas(10,11). Exsolution of dissolved volatiles (carbon dioxide and water) is thought to be essential to provide sufficient buoyancy for the rapid ascent of these dense, crystal-rich magmas. The cause and nature of such exsolution, however, remains elusive and is rarely specified(6,9). Here we use a series of high-temperature experiments to demonstrate a mechanism for the spontaneous, efficient and continuous production of this volatile phase. This mechanism requires parental melts of kimberlite to originate as carbonatite-like melts. In transit through the mantle lithosphere, these silica-undersaturated melts assimilate mantle minerals, especially orthopyroxene, driving the melt to more silicic compositions, and causing a marked drop in carbon dioxide solubility. The solubility drop manifests itself immediately in a continuous and vigorous exsolution of a fluid phase, thereby reducing magma density, increasing buoyancy, and driving the rapid and accelerating ascent of the increasingly kimberlitic magma. Our model provides an explanation for continuous ascent of magmas laden with high volumes of dense mantle cargo, an explanation for the chemical diversity of kimberlite, and a connection between kimberlites and cratons.