It is the first time that the ICJ, the UN's top court, has ruled that an event was genocide, which was made an international crime under a 1948 treaty.
However, in the same ruling the court has also said that the mass killings in other parts of Bosnia during the 1992-95 war did not constitute genocide.
"The court finds it has not been established by (Bosnia) that the killings amounted to genocide," Higgins said.
Bosnia has accused Serbia of masterminding the widespread "ethnic cleansing" of Bosnian Muslims and Croats during the 1992-95 war, saying it amounted to genocide.
If they had been successful, Serbia could have been liable for billions of dollars in compensation to Bosnia.
In its ruling, the court said that Serbia "was making considerable military and financial support available to the (Bosnian Serb entity) Republika Srpska" during the Bosnian war; but did not make any immediate comment about specific involvement in genocide.
More than 100,000 people died in the Balksans war between 1992 and 1995; it triggered the break-up of the former Yugoslavia.
Bosnia's Muslims and Croats wanted to cut all ties with the capital, Belgrade, a move that was opposed by Bosnian Serbs.