It was not immediately clear if an announcement will precede the execution.
A.K.M. SHamsul Islam, an official of the Supreme Court, said the justices on Thursday forwarded the court decision to authorities.
The developments come at a time of deep tension in Bangladesh, a nation struggling to overcome extreme poverty and rancorous politics.
Security officials opened fire to disperse opposition activists in eastern Bangladesh, leaving at least three people dead and 15 others wounded, Dhaka's leading Bengali-language Prothom Alo newspaper reported.
The chaos broke out in Laxmipur district, 96 kilometers (60 miles) east of Dhaka, during a nationwide opposition blockade after elite security forces raided and searched the home of an opposition leader, the daily reported.
Mollah would be the first person executed in special trials begun by Prime Minister Sheikh Hasina in 2010 of people suspected of crimes during the war of independence.
The government says Pakistani soldiers, aided by local collaborators, killed 3 million people and raped 200,000 women during the nine-month war.
Most of the defendants are opposition members. Mollah's party and the Bangladesh Nationalist Party say the trials are an attempt to weaken the opposition and eliminate Islamic parties.
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