In addition to its global North-South anisotropy, there are two other enigmatic seismological observations related to the Earth's inner core: asymmetry between its eastern and western hemispheres and the presence of a layer of reduced seismic velocity at the base of the outer core. This 250-km-thick layer has been interpreted as a stably stratified region of reduced composition in light elements. Here we show that this layer can be generated by simultaneous crystallization and melting at the surface of the inner core, and that a translational mode of thermal convection in the inner core can produce enough melting and crystallization on each hemisphere respectively for the dense layer to develop. The dynamical model we propose introduces a clear asymmetry between a melting and a crystallizing hemisphere which forms a basis for also explaining the East-West asymmetry. The present translation rate is found to be typically 100 million years for the inner core to be entirely renewed, which is one to two orders of magnitude faster than the growth rate of the inner core's radius. The resulting strong asymmetry of buoyancy flux caused by light elements is anticipated to have an impact on the dynamics of the outer core and on the geodynamo.