The Welsh international winger, who joined Los Blancos last night in a deal believed to be worth around £85m, was speaking at his unveiling in front of supporters at the Santiago Bernabeu this lunchtime.
"I just want to say that it is amazing to be here. I hope I can help bring success to the club and that we can win the club's 10th European Cup this season," he said.
The attacker also attempted to speak Spanish, saying: "Hello. It is a dream to be here in Madrid. Thank you for this welcome."
Bale is now expected to travel back to United Kingdom to link up with his Welsh teammates for international duty.