Grace, the world number 35, picked up eight birdies in a flawless third round to move to four under par, two shots behind leader Jordan Spieth.
There have been 31 rounds of 63 in major championships and Grace's feat prompted a standing ovation at the 18th green as he saluted the packed galleries at Royal Birkdale.
"I honestly had no idea that was something historic," Grace told reporters.
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"I knew I was playing well and I was just in the zone. I didn't know 62 was the lowest ever but now it makes it even more special. If my caddie knew he never said anything; good on him!"
American world number one Dustin Johnson also made a significant move up the leaderboard by collecting six birdies in his first 15 holes to get to three under.
Heavy rain and swirling crosswinds had made scoring difficult in the second round but the sun came out on Saturday morning and the world's top players took full advantage.
Local favourite Tommy Fleetwood, the world number 14, gave the home fans something to cheer about with a fine round of 66 and former world number one Jason Day of Australia shot 65 to move to level par.
Spieth, the world number three and twice major champion, will tee off in the final group out with fellow American Matt Kuchar who trails him by two shots.
England's Ian Poulter is in the penultimate group alongside American Brooke Koepka with both players on three under par.
(Editing by Clare Fallon)
