Birational motives, I: pure birational motives

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.