"A special day", Grace said.
Grace, ranked 35th in the world, made eight birdies in his bogey-free, eight-under-par third round at the par-70 Royal Birkdale course.
Thirty-one golfers have carded a 63 in a major championship, most recently Henrik Stenson in last year's Open Championship. And then Saturday at The Open, along came Branden Grace and an unprecedented 62.
"We knew what the score was, but I had no idea I was about to shoot the lowest round ever. I promise you", Grace added. "I was just grinding, grinding, and I just couldn't believe my final round, making so many birdies".
At 11 under par, Spieth enjoyed a three-shot lead over playing partner Matt Kuchar, with 20-year-old Canadian debutant Austin Connelly - a good friend of Spieth - three shots further back alongside US Open champion Brooks Koepka.
The 47-year-old American posted a 77, which left him 10 over after his first-round 73 and heading home for the first time in four outings at Southport. The world number 25 then two-putted from 50 feet on the 18th to finish a spectacular round and break a record that had stood since Johnny Miller's 63 at the US Open in 1973.
"Zack came up and said, 'You're in the history books, ' " Grace said.
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He was two shots behind Jordan Spieth, who was on the range still warming up, when Grace finished. He was one of many players to dive deep below par, and his performance was strong enough to vault him all the way into a tie for second at the Open Championship.
"He did it with his ball-striking".
He played the third round with Jason Dufner, who shot a 63 at Oak Hill in the second round of the PGA Championship four years ago.
"I didn't know what was going on on 18". Gary Player, who was only the third golfer in history to win the career grand slam after Ben Hogan and Gene Sarazen, won nine majors including three Claret Jugs. He shot 64 in the third round of the 2015 PGA Championship. His career low round in a PGA Tour-sanctioned event was a 63 at the WGC-HSBC Champions event previous year.
Grace will also be a member of the International Team, once again, at this year's Presidents Cup, a match play-style event against the United States which resembles the Ryder Cup - taking place at Liberty National Golf Club in New Jersey this September.
"I was hitting it well, couldn't have asked to get off to a better start", McIlroy said.
"My main thing was not wanting to finish with a bogey".