The will, and rebirth
Not according to U Dhammaratana's commentary on the Visuddhimaga... There it is said:


  It could be said that they [kamma-bhava and sankhara] are two
  different ways of sayin the same thing


Kamma-bhava is what recreates the next rebirth. Sankhara is kamma-bhava considered without the "associated mental states".

I think I see no way round this, besides questioning Buddhaghosa or the accuracy of the commentary. Unless sankhara is empty in a different way to kamma-bhava :/