Brazauskas announced Wednesday his coalition government would step down, after one of its largest parties, Labor, withdrew amid allegations of corruption.
Adamkus has 15 days in which to propose a new prime minister candidate to the Parliament.
Prime ministers typically oversee the government until a successor takes office, but Brazauskas, apparently stung by Adamkus' assertion that he couldn't trust the outgoing government, said his resignation would take effect immediately.
Adamkus is expected later Thursday to name Finance Minister Zigmantas Balcytis as acting prime minister until a replacement is voted into office, the AP reports.
Adamkus' nominee for the post will have 15 days in which to propose a Cabinet and government platform to the Parliament. A simple majority in the 141-seat assembly is needed to approve a new government.
Brazauskas' Social Democrats, with 23 seats in Parliament, are expected to have a key role in the talks.
The country of 3.5 million, which joined the EU and NATO in 2004, has grown accustomed to political upheaval since breaking free from the Soviet Union in 1991. The former president, Rolandas Paksas, was impeached in 2004. He was later convicted of leaking state secrets to a key financial backer.
Brazauskas' government, Lithuania's 13th in the 15 years since it regained its independence, collapsed as the Labor Party announced Wednesday it was leaving the coalition. It was the second party to withdraw from the ruling bloc, which originally consisted of four parties.
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