Let 0 be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph Ᏼ we define 0 (Ᏼ) to be the maximum size of a set of pairwise triple-disjoint copies of 0 in Ᏼ. We say a function from the set of copies of 0 in Ᏼ to [0, 1] is a fractional 0 -packing of Ᏼ if ¥ ʯe ( ) Յ 1 for every triple e of Ᏼ. Then * 0 (Ᏼ) is defined to be the maximum value offor all 3-uniform hypergraphs Ᏼ. This extends the analogous result for graphs, proved by Haxell and Rödl (2001), and requires a significant amount of new theory about regularity of 3-uniform hypergraphs. In particular, we prove a result that we call the Extension Theorem. This states that if a k-partite 3-uniform hypergraph is regular [in the sense of the hypergraph regularity lemma of Frankl and Rödl (2002)], then almost every triple is in about the same number of copies of K k (3) (the complete 3-uniform hypergraph with k vertices).