The 25-year-old Ghanaian international joins on a three-year contract after passing a medical and agreeing personal terms. He trained with his new team-mates for the first time this morning and will be in the squad for Saturday's Sky Bet Championship match at Charlton Athletic.
Boro Chief Executive Neil Bausor said: "We're absolutely delighted to have completed the signing of such a talented player who we believe can really excite our fans.
"Albert is a Ghanaian international who has hopes of representing his country at next summer's World Cup finals.
"We hope playing for Middlesbrough can help him achieve that ambition and at the same time he can be an important player for us and help us achieve our ambitions as a club."
Boro boss Tony Mowbray said he has been interested in signing Adomah for a couple of years.
"Albert is a creative wide player who can go up and down the right wing and cause teams problems with his speed and his trickery. He can create chances and he can also drop his shoulder and smash it in the net.
"He'll be a welcome addition for us. We're looking to strengthen other positions as well and we also need a player who can stick it in the net.
"He can add a little bit of excitement, thrust and forward speed to our team. Let's hope he can have a decent time with us."
London-born Adomah started his career with Isthmian Premier side Harrow Borough before joining League Two side Barnet, where he scored 19 goals in 112 appearances.
He moved to Bristol City in July 2010, with the initial fee of £150,000 being set by a tribunal, and his exciting style of play quickly made him a fans' favourite at Ashton Gate.
Adomah was named player of the year in 2010-11 and went on to make 131 appearances for the Robins, scoring 17 goals.
He has made nine appearances for Ghana, playing in this year's African Cup of Nations.
Adomah scored in both games as Bristol City recorded a double against Boro last season, a 3-1 win at the Riverside and a 2-0 victory at Ashton Gate.