Glover -- whose only PGA Tour victory came three years ago at the Walt Disney Classic -- has made the cut in seven of his nine appearances this year. Nevertheless, he has finished in the top 25 only twice and ranks 117th in the FedEx Cup points race.
Weekley, who chipped in at the final hole to win last year's tournament, shot a 64 over the Harbour Town Golf Links to finish 36 holes at 133. Anthony Kim was third at 134 while Stewart Cink, Stephen Ames and Cliff Kresge shared fourth at 135.
Cink finished third at last week's Masters.
Glover began play on the back nine and promptly bogeyed the par-4 10th. But he made six birdies in a nine-hole stretch beginning at the 15th.
"Obviously a good score, but a pretty slow start," Glover said. "I made my first real good swing of the day at the 15th and after that played pretty solid."
First-round co-leader and five-time Heritage champion Davis Love III settled for a 71 that left him tied for 14th at 137.