The 32-year-old completed his move to Anfield after undergoing a successful medical and agreeing personal terms on Monday.
He joins on a two-year contract with the option of an extra year.
#️⃣1️⃣3️⃣ pic.twitter.com/FMjV5Ymx7H— Liverpool FC (@LFC) August 5, 2019
"I think this is the perfect solution for us in this moment, replacing talent and experience like for like," Liverpool boss Jurgen Klopp said of Adrian "With Ali and Simon, I felt like we had two No. 1s and in many ways we have repeated that with Adrian.
"This is important. He is a 'keeper who knows the Premier League, who has played in pressure situations and has the highest quality.
"They are big gloves to fill because Simon was such an outstanding professional and contributed to the group even when not in the starting XI. Everything we know about Adrian tells us he has the personal qualities to go with his clear playing talent to ensure this area of the team is not weakened.
"We're all looking forward to working with him."
Having left West Ham this summer, Adrian was due to hold talks with Real Valladolid, but stopped those negotiations once Liverpool made him an offer.
Klopp was after an experienced No. 2 to fill the void of Mignolet's exit and the Spaniard ticks that box.
His six years with the Hammers saw him make 150 appearances in all competitions, with a professional career total of 287 outings.
Adrian is expected to be in the matchday squad on Friday when Liverpool kick off the new Premier League season against Norwich City at Anfield.