Cloning of modern mammals is done using intact cells. One of the big obstacles to cloning mammals from remains like frozen mammoth cells is that the cells are never whole. They burst from the freezing of the water inside them.
Japanese scientists, though, have now gotten around that, successfully cloning live mice from cells frozen for 16 years. It's still quite a jump to cloning mammoth cells frozen thousands of years ago (we still need a surrogate mother, presumably an elephant, and it's not clear elephant and mammoth are close enough for this to work), but the idea is edging into the realm of possibility.