And Zack [Rasego, Grace's caddie] came up and said: 'You're in the history books.' And I was like: "'What are you talking about?'"
Describing what the achievement meant to him, Grace said: "It's special".
According to Grace, he wasn't even aware of his record-setting round as it was happening. Grace was only thinking about trying to get through the third round without a bogey. I had no idea that 62 was the lowest ever. Perfectly-manicured surfaces. A course, already short at 7,156 yards, shortened to 7,027 for the third round, with the par-4 fifth reduced 36 yards to 310 to make it more driveable, and the par-3 seventh some 20 yards shorter at 156 yards.
Asked what he took out of his viewing, Spieth said "That Royal Birkdale notoriously hard had just become one of the easier golf courses that we play for one round for the year". With clear skies and little wind - a significant departure from the windy and wet conditions during Friday's second round - Grace was able to hit 16 of 18 greens and sink eight birdies with no bogeys.
He walked up to the par-4 18th needing just a par for his 62.
When Justin Thomas shot the last 63 at a major, at the U.S. Open a month ago, Miller was immediately sought out for comment.
But Rory McIlroy's chances of landing his first major since the 2014 PGA Championship look to be over after he paid the price for a number of careless mistakes following another bright start to his round. The lowest one at the end of play on Sunday will end up with the claret jug regardless.
World number two Hideki Matsuyama of Japan carded 66 to join Grace at four under, one ahead of American world number one Dustin Johnson (64), defending champion Henrik Stenson (65), Spain's Rafael Cabrera Bello (67) and American Chan Kim (67). Kuchar missed the fairway, couldn't find the green with his second and took four more to get down.
Spieth won The Masters and US Open in 2015 and will be looking to complete the third leg of a career Grand Slam today after a wonderful third round, during which he went head to head with Kuchar for 18 holes. "There's a lot of spots you want to keep out of on this golf course".
And there were plenty of low scores offered on this day at Royal Birkdale. "So just fortunate the way things finished".
Grace - playing in the 13th pairing alongside Jason Dufner - went out in 29 to set up his bid for history.
He made birdie at each of his last four holes. "I'm not going to take anything away from a guy shooting 63 on a par 72".