Inspector General of Prisons Mainuddin Khandaker had said Mollah, of the opposition Jamaat-e-Islami party, would be hanged shortly after midnight at Dhaka's Central Jail.
Deadly clashes have followed court verdicts against six other current and former officials of the Islamic party, an ally of the main opposition Bangladesh Nationalist Party, and extra police were stationed in the capital to head off any violence. Paramilitary guards were on standby across the country as well.
Mollah's party and the Bangladesh Nationalist Party say the trials are politically motivated in an attempt to weaken the opposition. International human rights groups have raised questions about the impartiality of the tribunal. Authorities have denied the allegations.
On Monday, New York-based Human Rights Watch urged the government to halt Mollah's execution.
Mollah was found guilty by the special tribunal in February and sentenced to life in prison. The Supreme Court then changed the penalty to a death sentence in September, triggering deadly clashes and a nationwide general strike.
Junior Law Minister Quamrul islam said prison authorities read the death warrant to Mollah on Tuesday afternoon and asked him if he wished to seek presidential clemency but he did not seek that.