The 37-year-old was severely injured in a gun accident in 1997 in which he lost his nose and lips and was left with limited mouth movement.
Mr Norris underwent multiple life-saving and reconstructive operations. However, he has spent the past 15 years living as a recluse and wore a mask on the rare occasions he ventured outside.
The new face came from an anonymous donor whose organs were also used to save the lives of five other people. Four of the organ transplants took place over the course of two days at the hospital.
US doctors have carried out what they say is "the most extensive full face transplant" to date