Kim Jong Il, who had ruled the country since 1994, died at age 69 of "great mental and physical strain" Saturday while traveling in a train, the state-run Korean Central News Agency reported.
The report said the "Korean revolution" is now led by Kim Jong Un.
"All party [Workers' Party of Korea] members, service persons and people should remain loyal to the guidance of respected Kim Jong Un and firmly protect and further cement the single-minded unity of the party, the army and the people," a notice published by the news agency said.
CNN reported the party had dubbed the son as the "Great Successor."
China's Xinhua news agency quoted the notice as urging the nation to turn sorrow into strength and courage under the son's leadership.
"We should increase the country's military capability in every way to reliably safeguard the Korean socialist system and the gains of revolution," the notice said.
Kim Jong Un, believed to be in his late 20s, is the youngest son of the late leader.
Last year, Kim Jong Un became a four-star general, CNN said.
He is believed to speak some English and German.
The death of Kim Jong Il has raised anxiety among diplomats, military strategists and political leaders as to what awaits North Korea.