Stallings' 12-foot putt on the par-4 18th caught the right edge and dropped in for the second 60 in two days at Grand National's rain-softened Lake Course.
Stalling had the lowest round of his PGA Tour career and broke the tournament 54-hole record at 19-under 194. He won the last of his three tour titles in 2014.
Grayson Murray was second after his second straight 64.
Collins followed his 60 with a 69 to drop into a tie for third with Tag Ridings (63) at 17 under.
Rory Sabbatini shot a 62 to jump from 54th to a tie for 14th at 11 under.
Jim Furyk, the only player to shoot two sub-60 rounds in PGA Tour history, had his second straight 68 to get to 8 under. The 47-year-old U.S. Ryder Cup captain is playing the event after failing to qualify for the British Open.