The 21-year-old attacker has signed a four-year deal, plus a club option, for an undisclosed fee, believed to be a £4.3m package.
The Senegal international, who will wear the number 19 shirt, attracted firm interest from Championship rivals Watford, but his preference was a summer switch to Carrow Road.
Diallo’s move is now subject to the standard Football League ratification and a work permit, which is expected to be a formality.
“I’m so excited for this new journey. The sporting director and head coach have been very important for me and I can’t wait to start working at the club,” said Diallo, speaking to the club’s official site.
“It has been my dream to play in England – something new for me. But I am ready and prepared for the challenge ahead.
“I really always wanted Norwich City. I am excited about the project and the vision for this club and I can’t wait to come to England and meet my new team-mates and the supporters.”
City officials met Diallo in Paris on Tuesday to being his visa process and undergo a medical, and the right footer becomes the ninth senior signing of a busy summer led by sporting director Ben Knapper and head of recruitment, Lee Dunn.
“Papa is a huge talent, and we’re delighted to have won the race for his signature,” added Knapper, on club channels.
“Despite having multiple offers from across Europe, he fought hard to join us, and we’re excited to have him on board to help us work towards our objective of reaching the Premier League.
“He’s a player with enormous potential and he’ll add pace, and a real directness, to our attacking options. He can play in multiple positions and is comfortable operating on both the outside and inside.
“He will provide real penetration both with his movement and ability to carry the ball and commit players one versus one.
“He’s had a great footballing education up until this point and we’re looking forward to seeing him further develop in our environment as both a person and player.
“We’re delighted to welcome Papa and his family to Norwich City.”
Jeffrey Schlupp was confirmed earlier on Thursday, and both new signings are expected to link up with Liam Manning’s squad when they return to England.
City were in the market for reinforcements across the frontline following the recent departure of Borja Sainz to Porto in a £14.25m transfer.
Diallo can operate on either flank, as well as down the middle, and produced 11 goal contributions in 28 Ligue 2 appearances to help earn Metz promotion last season.
But the Ligue 1 newcomers reportedly need to raise €20m in player sales ahead of their return to the French top flight due to uncertainty around domestic broadcast deals.
Diallo only had 12 months left on his current contract, and along with teenage striker Idrissa Gueye was viewed as Metz’s most saleable assets.
Manning’s current squad complete their 10-day tour of the Netherlands with a friendly double header on Friday against Olympiacos and ADO Den Haag.