The 24-year-old has agreed a four-year deal with the club, having passed a medical at the American Express Elite Football Performance Centre earlier today.
Head coach Graham Potter said, “Adam is a player the club has been aware of for sometime, and someone we came up against last season at Swansea City.
“He is a quality defender, comfortable on the ball, fits the profile of the type of player we want to bring to the club, and he was very keen to join us.
“He is still relatively young, but has good experience from the number of games he has played. Predominately a centre-half, he can also play at right-back, and he will bring additional competition alongside our existing defensive options.”
Born in Chichester, West Sussex, Webster began his career just across the Hampshire border with Portsmouth; before moving to Ipswich Town in 2016, with Albion defender Matt Clarke moving in the opposite direction as part of the deal.
Webster has made more than 200 senior appearances, with around half at Championship level. He played 47 times last season for City, after joining from Ipswich a year ago, scoring three goals.
He has also represented England at under-18 and under-19 level, where he will have worked alongside Albion’s technical director, Dan Ashworth.