Rangers boss Stuart McCall says he feels sorry for Gael Bigirimana, but admits he should never have been signed.
Chris Gunter will stay at Reading, after the club took up the option to extend his deal by another year.
The 28-year-old Yorkshireman has signed a three-year deal with the Seagulls and becomes their fourth new recruit of the close season, following Aaron Hughes, Nzuzi Toko and Chris O’Grady.
Stockdale began his career with York City and then joined Darlington before moving to Fulham in 2008 in a deal reported to be worth £600,000.
He has since had loan spells with Rotherham, Leicester, Plymouth, Ipswich and Hull, helping the Tigers secure promotion to the Premier League in 2012/13.
He made a total of 51 senior appearances for Fulham and established himself as first-choice keeper at Craven Cottage last season.
Albion manager Sami Hyypia said: “We are delighted to have completed David’s signing from Fulham, and I am pleased to welcome him to the club.
"It is a transfer we have been working on for some time - and David is a goalkeeper with excellent pedigree. He has got very good experience of the Premier League and Championship, but at 28 is in his prime."
Brighton's goalkeeping coach Antti Niemi played alongside Stockdale at Fulham, and Hyypia added: "Antti knows David very well from their time together as players at Fulham. He knows his strengths and attributes, and David was on the fringes of the England senior squad not too long ago.
"Alongside Casper Ankergren and Christian Walton, David’s arrival means we are satisfied with our goalkeeping options ahead of the new season.
"We still have some work to do in other areas in terms of recruitment, but we are making good progress and this is an excellent signing and another quality addition to the squad."