Fix a prime number l. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goneharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data). © 2003 Kluwer Academic Publishers.