I will be reincarnated in accordance with the laws of conservation of mass and conservation of energy.
Before you can have a discussion about reincarnation, you need to define a soul.
That being said, reincarnation on atomic level is a true phenomenon (e.g. iron atoms in your body were born in supernova(s)).
To believe in reincarnation, you pretty much have to believe in spirits (souls, ghosts). I would expect that anyone who doesn't believe in the big spirit in the sky doesn't believe in ghosts, spirits, or souls. They're kind of necessary for reincarnation, as I understand it.