Fearn et al. (1981) recently proposed a mechanism whereby the inner core contains a significant fraction of liquid. We construct a model of a partially molten inner core compatible with seismic observation. Specifically, we develop an improved method of bulk anelasticity including the effect of thermal and material diffusion and show that it is quantitatively capable of explaining the observed anelasticity. Also, studies of the reflection of seismic waves from the inner-core boundary are used to estimate the mass fraction of liquid in the inner core.