Florida (11-7, 4-4) snapped a 10-game losing streak to Kentucky and won in Rupp Arena for the first time since Feb. 12, 1989.
Kentucky (19-3, 8-1), which leads the series 72-21, also had a nine-game winning streak come to an end.
Not only did Florida shoot a season-high 57 percent (12-of-21) from 3-point range, the Gators also beat Kentucky at its strength in taking a 40-30 advantage in rebounds despite foul trouble by big men Damen Maddox and Greg Stolt.
Williams, who made his first seven shots, finished with 8-of-10 shooting from the field. He hit 4-for-5 of his 3-point attempts and handed out four assists. Weaks hit 7-of-14 shots.
Kentucky's Cameron Mills also turned in a scintillating shooting performance, hitting 8-of-14 shots from beyond the 3-point arc in finishing with a career-high 31 points.
Florida trailed only once in the game, 2-0 after Kentucky's Nazr Mohammed's dunk to start the game.
After that, Williams hit five straight shots, including three 3-pointers, as Florida went up 31-16 with 7:44 remaining in the half.
Kentucky battled back behind Mills' 14 points to trail 40-34 at the half but the Gators played with poise in the final 20 minutes to turn back all challenges mounted by the Wildcats.
Wayne Turner's 3-pointer after a Florida turnover sliced the margin to 47-45 with 16:57 to go but Kentucky would draw no closer as Weaks took charge for the Gators.
Weaks scored on a short jumper, hit two free throws and drilled two 3-pointers in a 10-4 run to put Florida up 59-49 at 12:26.
Kentucky got no closer than 82-75 on Mills' 3-pointer with 1:20 to go. Florida went without a field goal in the final 5:30 but made 12-of-18 free throws down the stretch.
Dan Williams finished with 12 points, Eddie Shannon had 11 and Stolt 10 for Florida, which shot 50 percent (27-of-54) from the field.
Scott Padgett had 10 points for Kentucky, which made 43 percent (27-of-63) of its shots. The Wildcats made 12-of-32 shots from 3-point range.
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