  A famous example of Casson and Gordon shows that a Haken 3-manifold can have
an infinite family of irreducible Heegaard splittings with different genera. In
this paper, we prove that a closed non-Haken 3-manifold has only finitely many
irreducible Heegaard splittings, up to isotopy. This is much stronger than the
Waldhausen conjecture. Another immediate corollary is that for any irreducible
non-Haken 3-manifold M, there is a number N, such that any two Heegaard
splittings of M are equivalent after at most N stabilizations.
