The 29-year-old South Africa has now put himself into contention at the #Open Championship at Royal Birkdale with his third round.
Make no mistake, Grace knew exactly what was at stake as he set out on the third round of The Open.
Spieth had five birdies, including one on the final hole to which the crowd responded with a huge roar.
Spieth shot a 65, however, and the day ended with Grace seven shots out of the lead. He stalled slightly on at the start of the back nine with four pars in a row but kicked into gear again, with a 35ft birdie putt on the par-3 14th and another on the 16th. Playing on a wet and receptive course without much wind in the air, scoring conditions were ripe today.
Grace even admitted in his post round interview that he had no idea of the history he had just made, "I didn't know on 18 what was going on".
A birdie on No. 1 got them going. I was so in the zone and playing the round so well. The tap-in was a formality, and the tens of thousands of people who had poured into this magnificent venue had seen history made.
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This was Grace's second time flirting with the magical number. But he will settle for his day's work.
He's never finished better than third in a major tournament, and his best result in an Open Championship is a tie for 20th two years ago at historic St Andrews. Indeed, Greg Norman and Vijay Singh each did it twice.
Others did know. The 2015 PGA Championship victor Jason Day, having played five groups before Grace, said, "I think this is a good formula to see maybe the first 62 in a major championship".
Numerous game's biggest stars had near-misses. He could join Jack Nicklaus as the only players to win three major championships by the age of 23. Woods later referred to his score as "62-and-a-half". He had a 10-foot birdie putt for 62 in the second round of the 2013 PGA Championship at Oak Hill that he left short. Sometimes it helps not knowing these things. And then Saturday at The Open, along came Branden Grace and an unprecedented 62. "I hadn't a clue about the significance of the round until Jason's caddie informed me", he said.
Tommy Fleetwood's romantic adventure at the Open championship took another "lovely" upward turn on Saturday as the local hero shot a four-under-par 66 to leave him still dreaming of a storybook finish to his remarkable week.