On Wednesday, schools and businesses were closed as the strike was enforced. Police fired tear gas to disperse opposition supporters who exploded homemade bombs, barricaded roads and threw stones at security officials in some towns.
Mollah was previously convicted by a special war crimes tribunal in February and sentenced to life in prison. Both the defense and prosecution appealed that sentence to the Supreme Court.
Attorney-General Mahbubey Alam said Tuesday's verdict was now final, with no option for another appeal through the courts. He said Mollah's family can seek presidential clemency.
Defense counsel Abdur Razzaq said they were "stunned" by the court's decision to increase the sentence to death.
The ruling Awami League and its allies welcomed the verdict.
Mollah's party is an ally of the country's main opposition Bangladesh Nationalist Party headed by former Prime Minister Khaleda Zia, a rival of current Prime Minister Sheikh Hasina.
Hasina formed the special tribunal in 2010 to try war crimes suspects. Bangladesh says Pakistani soldiers, aided by local collaborators, killed 3 million people and raped 200,000 women during the nine-month war.
Zia has accused the government of using trials to weaken the opposition. The government denies the allegation and says it won power in 2008 with an election pledge to prosecute war crimes suspects.
Several other top leaders of Jamaat-e-Islami have been convicted of similar charges.
The government says the trials are being held at an international standard, but New York-based Human Rights Watch has raised questions about the impartiality of the tribunal.
The earlier sentence against Mollah also led to protests across the country, both by his supporters and those who said the sentence was too lenient.