The deal for Cucurella to join the Blues is worth more than $60 million.
In his only season with the Seagulls, the left-back scored one goal in 38 games.
I'm very happy." Cucurella said it was a big chance for him to join one of the best clubs.
I want to be happy here and help the team.
It is the most expensive sale in the history of the company.
Cucurella won a senior cap for Spain and was part of the team that won a silver medal at the Tokyo Olympics.
Cucurella handed in a transfer request after City expressed an interest in him.
"We did not want to lose him, but he wanted the opportunity of playing European football with a view to speeding up his progression to Spain's World Cup squad," said Graham Potter.
He had an excellent first season in the premier league, winning both our player of the season awards, and has been an absolute pleasure to work with over the past year.
Cucurella is the second big-money departure from the Seagulls this summer.
Todd Boehly is the chairman of the club.
"We continue to work on and off the pitch, and we're thrilled that he's going to be a part of the future at the club."
In the summer of this year, they have brought in England forward Sterling from Manchester City for £50m, a defender from Napoli for $34m, and a player from Villa for 20m.
They also signed 18-year-old American goalkeeper Gabriel Slonina for a reported £12m from Chicago Fire, where he will remain on loan until the end of the Major League Soccer season.
Gary Neville, a former Manchester United defender, made comments about Boehly on Friday as he was previewing the opening game of the English premier league.
Neville said Boehly, who is also interim sporting director "looks like he wants to play Football Manager" and described his transfer approach as "panicky".
Tuchel asked, "Panicking?" I would describe it as a lot of hard work and learning.
It's pretty much going on, new ownership and then stepping into the transfer market is probably one of the toughest things you can do when you're outside.
I have nothing but nice things to say about both Todd and Behdad.
The transfers so far speak for themselves and we have a very honest, very open, very direct line of communication.
The delay was caused by the sanctions and the loss of key players.
Both of the owners are involved in the daily business and the transfer market speaks for itself.
Levi Col will be going on a season-long loan from the Seagulls.
In the Championship play-off final, the England Under-19 international scored an own goal, but it was not enough as the Terriers lost to the Forest.
Potter saidLevi has enormous potential and has already shown his ability last season in the Championship and with England at various age group levels up to under-21s.
He plays mostly at centre-back, but has also played on the left as a full-back.
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