The invention relates to a method of splitting apart a substrate of two adjoining wafers defining between them a cleavage plane, by bringing each substrate into a substrate-receiving space; and clamping first and second jaw portions onto each substrate in such a manner as to hold each substrate and urge apart the two wafers of each substrate by co-operation between the shapes of housings in first and second portions of the two jaws, respectively. The invention also relates to a splitting method that includes bringing each substrate into a substrate-reception space; clamping together separator portions onto each substrate so as to split apart the two wafers of each substrate; and clamping the split-apart substrate wafers so as to hold the wafers together. An automated system for processing multiple substrates is also provided.