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Robert F. Bauer, chief counsel for President Obama’s re-election campaign, told a University of Maryland forum last week that he doesn’t think the new state voter-ID laws will have much impact on voter turnout in the fall, saving that Democrats have made “significant progress” in blunting the effects of the new laws.

But Mr. Larsen said his bill is designed to protect the integrity of elections over the long term.

“The effort isn’t done to counter these [state] laws,” he said. “These laws apply to elections in the future, and if we let them stand I believe you’ll see the steady erosion of voting.”