Cunha, who is often compared to the dark, manipulative hero Frank Underwood of the hit Netflix series "House of Cards," used his position as speaker of the lower house to put Rousseff on a so-far unstoppable path to being forced from office.
A conservative lawmaker allied with Congress's Evangelical Christian wing, Cunha is accused of taking millions of dollars in bribes and maintaining secret Swiss bank accounts.
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"I am paying a high price for having launched the impeachment," he said.
Cunha also said that by resigning he'd bring stability to Congress, where he is a polarising figure and where until now he has used his legendary wheeler-dealer skills to fend off attempts at pushing him out.
"The chamber cannot put up with this indefinitely," he said.
Cunha was already suspended from his powerful post in May, but he retained considerable support from a network of political allies built during a career in which even critics have recognised his prowess as a backroom operator.
His downfall is not expected to change the fate of Rousseff, from the leftist Workers' Party, which has been in power for 13 years but is now in grave danger.
Not only does Rousseff face ejection from office but Workers' Party founder and former President Luiz Inacio Lula da Silva is embroiled in serious corruption allegations of his own.
Thanks largely to Cunha's leadership, the lower house voted heavily in favor last year for opening an impeachment trial against Rousseff, on charges that she illegally manipulated government accounts. She describes the process against her as a coup in disguise and has called Cunha the lead plotter, along with Michel Temer, her former vice president who is now temporarily in her job.
Both Cunha and Temer are from the centrist PMDB party, the biggest in Brazil and formerly in a shaky alliance with the Workers' Party.
In May, the Senate followed the lead of Cunha's lower house and voted by a large margin to launch the impeachment trial. Senators are now due in late August to take a final vote on whether to terminate Rousseff's presidency.
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