  This is a considerably expanded version of the "pure" part of our 2002
preprint. We define a category of pure birational motives over a field,
depending on the choice of an adequate equivalence relation on algebraic
cycles. It is obtained by "killing" the Lefschetz motive in the corresponding
category of effective motives. For rational equivalence, it encompasses Bloch's
decomposition of the diagonal. We study the induced Chow-K\"unneth
decompositions in this category, and establish relationships with Rost's cycle
modules and the Albanese functor for smooth projective varieties.
