Areola has cost West Ham £7.75m and has signed a five-year contract with the option to extend for a further year.
The 29-year-old spent last season on loan at the London Stadium, making 18 appearances across all competitions, albeit just one of those came in the Premier League.
Areola has also had loan spells at Villarreal, Real Madrid and Fulham, and was one of France’s reserve goalkeepers when they won the FIFA World Cup in 2018.
Speaking to club's official website about his arrival on a permanent basis, Areola said: "I feel good, I feel great, I'm happy to sign and happy to stay here permanently and I can't wait to start to train and to do the job.
"The main thing is that I felt the love of the supporters last season. Something for me that is really important is to feel that I'm loved. I know that I have to do my job and to do everything to make them happy, and obviously also the team, my teammates, the staff and Xavi Valero as well. The vibe of last season was great, so I just wanted to do everything to stay here.
"I feel comfortable, I feel like when I'm signing in a club, I want to feel that it's like my second family. Obviously, we're here every day and we are sharing lots of times together, so I have to feel comfortable with everyone and feel them comfortable with me as well, so I just feel comfortable here."
Manager David Moyes added: "Alphonse made a huge contribution to the club during his loan spell last season. His performances backed up why we see him as a top goalkeeper.
"He brings proven quality and experience at the highest level, both on the domestic and international scene. We're delighted he's with us permanently and look forward to welcoming him back for pre-season."
Areola will compete with Lukasz Fabianski for the number one spot next season, which begins with a home game against champions Manchester City.
