Hundreds of pro-government activists, meanwhile, blocked traffic on a main road in Dhaka demanding Mollah's immediate execution.
The developments come at a time of deep tension in Bangladesh, a nation struggling to overcome extreme poverty and rancorous politics.
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 nation's war of independence against Pakistan in 1971. 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 in the trials 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. International human rights groups have also raised questions about the impartiality of the tribunal. Authorities have denied the allegations.
Deadly clashes have followed court verdicts against six other current and former officials of Mollah's party, and extra police are stationed in the capital to head off any new violence. Paramilitary guards are on standby across the country as well.
Carrying out the execution would complicate an already critical political situation in Bangladesh, where the opposition has carried out violent protests for weeks to back a demand for an independent caretaker government to oversee the general elections early next year.
The government has rejected that demand and said a political government headed by Hasina would conduct the elections.
U.N. Secretary-General Ban Ki-moon spoke by phone with Hasina "to express his strong concern about the prevailing situation in the country," a statement from his office said Wednesday.
It said Ban urged Hasina to "resolve differences" over the upcoming elections with dialogue.
The election is set for Jan. 5, but the opposition alliance led by former Prime Minister Khaleda Zia plans to boycott it. Weeks of blockades and general strikes have left nearly 100 people dead since October. Mollah's party has been banned by the Election Commission from taking part in the elections.
Mollah's family had met him at a Dhaka jail on Tuesday for what was expected to be the last time.
As authorities finalized the time for the execution, many cellphone users in Bangladesh received text messages from an unknown number that said if Mollah was executed a civil war could break out. The Bangladesh Telecommunication Regulatory Commission said it is trying to determine who sent the messages.
Defense counsel Sazzad Ali Chowdhury said the postponement late Tuesday gave lawyers time to file the petition which the Supreme Court's Appellate Division was reviewing Wednesday.
Mollah was found guilty by the special tribunal in February of killing a student and a family of 11 and of aiding Pakistani troops in killing 369 others during the independence war. He was sentenced to life in prison. The Supreme Court changed the penalty to a death sentence in September, triggering deadly clashes and a nationwide general strike.
Until it gained independence in 1971, Bangladesh was the eastern wing of Pakistan. Mollah's party campaigned against Bangladesh's independence and has been accused of forming several groups to help Pakistani troops in killing, rape and arson.
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Associated Press writer Cara Anna at the United Nations contributed.
