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The bill includes a new definition of “persistent, extremely low minority turnout.”

Aides said the point was to create a separate test that could be used to spot potential discrimination in localities that weren’t big enough to have multiple adverse court rulings.

Other parts of the Voting Rights Act, which prohibit discrimination against any voter no matter the race or ethnicity, remain in place.

Subdivisions would be subject to preclearance if they are shown to have persistent low minority turnout and had one adverse judicial ruling in 15 years, or if they had three adverse court rulings. A state would be subject to preclearance if it had five documented rulings, at least one of which must be a state law.

States and subdivisions would be covered for 10 years from the latest violation.

The six jurisdictions that will be covered immediately is fewer than the number as of last year, when the Supreme Court invalidated the old formula. At that time, nine states and dozens of localities in seven others were covered.

Mr. Sensenbrenner and his fellow bill authors, Mr. Leahy and Rep. John Conyers Jr. of Michigan, the ranking Democrat on the House Judiciary Committee, said they compromised in order to make the bill palatable to a broad swath in Congress.

One of those was that the bill won’t count rulings on voter identification legislation. Those laws have proved to be popular, particular among Republican-led states, though federal judges have halted some of them.