Kim defeated Germany's Ole Bischof after losing to him in Beijing.
Bischof scored a semifinal victory over American Travis Stevens, who then lost to Canadian Antoine Valois-Fortier in one of the bronze medal bouts.
The other bronze medal went to Ivan Nifontov of Russia.
In the women's 63-kilogram class, Urska Zolnir won her second Olympic medal and first gold one with a decision over Xu Lili of China.
Zolnir also won the bronze medal at the 2004 Olympics in Athens.
The bronze medals awarded on Tuesday went to No. 1 seed Yoshie Ueno of Japan and to current European champion Gevrise Emane of France. Ueno had to come through the repechage round to gain a spot in a bronze-medal match.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys