Starting the day in a share of 20th at three-under, Ko looked to have shaken off the cobwebs from the back nine of her second round which included three bogeys.
She picked up birdies on the two par fives on the front nine and another at the par five 11th along with a fourth birdie at the 12th to move to seven-under and the top five - three shots from the leaders.
But her round wobbled with a double at the 14th and a bogey two holes later. Ko finished with a birdie at the 18th to be four-under across the par fives today.
Kupcho, meanwhile, started fast and kept on going.
"Everything was working," Kupcho said. "I mean, seriously, this week I think my putting is definitely the props. I have putted really well and you got to make putts in a major championship."
Kupcho moved into position to win for the first time on the LPGA Tour by birdieing eight of the first 12 holes in mostly calm conditions.
"Honestly, it's all a blur," Kupcho said. "I hit the fairways, hit the greens and really was just trying to put smooth putting strokes on them. That's what I did, and they fell."
As second-round leader Hinako Shibuno had a 77 to drop 12 strokes back, Kupcho flew past her, breaking the 54-hole record of 14-under set by Pernilla Lindberg in 2018 and matched by Tavatanakit last year. Dottie Pepper set the 72-hole mark of 19-under 269 in 1999.
Kupcho also matched her lowest tour score, hitting all but one fairway and three greens in regulation on the tree-lined layout.
"It's really special, just to be out here," Kupcho said. "I love this place. You just get positive vibes. It's such a beautiful course, so I think just taking it all in. But then again, just once it's time to hit a shot, focusing on that shot."
- With AP