Madrid announced Sunday that Bale has signed a six-year contract, and a person familiar with the deal said the fee was a world-record 100 million euros ($132 million). The person spoke on condition of anonymity because financial details are not being disclosed.
"I am not sure there is ever a good time to leave a club where I felt settled and was playing the best football of my career to date," Bale said in a statement published on the Tottenham website. "I know many players talk of their desire to join the club of their boyhood dreams, but I can honestly say, this is my dream come true."
Bale will be paraded at Madrid's stadium on Monday after undergoing a medical examination.
Tottenham manager Andre Villas-Boas said earlier in the week that Bale's move was set to be "the biggest transfer in world football," and the fee eclipses the €93 million Madrid paid Manchester United for Cristiano Ronaldo in 2009.
Madrid, whose annual revenue reaches €500 million, previously broke the transfer record in 2009 when Kaka joined from AC Milan for 65 million euros ($92 million).
All three blockbuster deals came under Florentino Perez, the club president who built the "Galacticos" team that included David Beckham, Zinedine Zidane and Figo.
The move for Bale, who had around three years remaining on his Tottenham contract, caps a rapid rise from White Hart Lane misfit to one of European football's most exciting players.
The 24-year-old Welsh winger has dazzled audiences around the world with swerving free kicks, defence-splitting surges and mesmerizing footwork.
But Bale was frustrated that Tottenham only qualified once for the Champions League, the stage where the world took notice of his talents for the first time in the 2010-11 season.
Now he's joining a club that has won the European Cup nine times.
"I am well aware that I would not be at the level I am today were it not for firstly Southampton and then Spurs standing by me during some of the tougher times and affording me the environment and support they have," Bale said. "Tottenham will always be in my heart and I'm sure that this season will be a successful one for them. I am now looking forward to the next exciting chapter in my life, playing football for Real Madrid."
Bale was signed by Madrid despite not winning a single title in his playing career.
What he does have is a collection of personal honours — and long highlight reel of spectacular goals.
In April he won the top two awards in English football after being voted player of the year and the young player of the year by his fellow professionals in English football.
The last player to achieve the Professional Footballers' Association double was Ronaldo in 2007, and the pair will now be on the same side in Spain.
In Bale and Ronaldo, Madrid manager Carlo Ancelotti will have one of the most formidable attacks in football to compete with Spanish champion Barcelona, which boasts Lionel Messi and new recruit Neymar.
The recruitment of Bale is Madrid's response to missing out on the 57 million-euro Neymar.
But the move was dragged out by Spurs wanting to spend the Bale windfall before reluctantly selling their main asset, and was only completed a day before the summer transfer window closes.
"Gareth was a player we had absolutely no intention of selling as we look to build for the future," Tottenham chairman Daniel Levy said. "He is a player whose career we have fostered and developed and he was only a year into his new four-year contract.
"Such has been the attention from Real Madrid and so great is Gareth's desire to join them, that we have taken the view that the player will not be sufficiently committed to our campaign in the current season."
Bale joined Tottenham in 2007 from Southampton for 5 million pounds (then $8 million) but endured a frustrating start to his Spurs career.
Although he scored three times in his first four starts, a series of injuries stymied his progress and Tottenham failed to win any of the first 24 Premier League games he appeared in for the club. That made him a symbol of poor results for fans until finally being part of a win on his 25th attempt, when he emerged as a substitute after 84 minutes.
Since then, Bale has never looked back. Neither has Tottenham, which at one point was reported to have tried to use Bale as a make-weight in deals to bring in new players.
After shaking off his substitute's role in January 2010, Bale began to consistently show the flair and pace that made him the first-choice left back.
And, as the attacking side of Bale's game began to flourish, he was pushed forward to the left wing.
Having helped steer Tottenham into the Champions League for the first time in May 2010, Bale was rewarded with a lucrative new deal — the first of three successive annual contract upgrades.
Bale announced himself to the world with two dazzling displays against Champions League holder Inter Milan in the group stage in the 2010-11 season, including a hat trick at the San Siro.
Bale's importance to the London club was underscored last season by the fact his 21 league goals helped win 22 points for Spurs although they still finished a place outside the top four.
And Bale had outgrown the team.
"I have had six very happy years at Tottenham but it's the right time to say goodbye," Bale said. "We've had some special times together over the years and I've loved every minute of it."
Joseph Wilson in Barcelona, Spain, contributed to this report.