Cunha, a key opponent of President Dilma Rousseff and architect of impeachment proceedings against her, "used his office for his own illicit ends to prevent the success of investigations against him," prosecutors said in asking the court to take action.
In March, the Supreme Court voted to put Cunha on trial, accusing him of taking USD 5 million in bribes as part of a vast embezzlement and bribery network centered on state oil giant Petrobras.
But the speaker -- who under Brazil's constitution is third in line to the presidency -- had kept ahead of his own enemies and remained the go-to man in the push against the president.
Senators will vote next week on whether to open an impeachment trial against Rousseff.
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Judge Teori Zavaski, in his ruling seen by AFP, said Cunha did not meet the requirements to exercise his role as speaker and was even less qualified to fill in for Rousseff.
The suspension comes at a time of deep political and economic crisis in the South American country, which will host the Olympic Games in August.