It is shown that surface tension can cause redistribution of melt in a partially molten medium, in accordance with a non-linear diffusion equation for the melt fraction. The associated diffusivity D depends on the surface energy and its positive (stable) for dihedral angle 60¿. In the more likely stable case. D~ 10-7 cm2 s-1 is typical for mantle melts, but a value as high as 10-1 cm2 s-1 is conceivable for volatile-rich fluids. Surface tension may play an important role in creating pathways for metasomatizing fluids in the Earth, but does not appear likely to affect substantially the existing estimates of large scale melt migration (e.g., beneath mid-ocean ridges).