Voters pushed back decisively after the Supreme Court overturned Roe v. Wade last year, approving ballot measures that established or upheld abortion rights in all six states where they appeared.

Now, with abortion rights groups pushing for similar citizen-led ballot initiatives in at least six other states, Republican-controlled legislatures and anti-abortion groups are trying to stay one step ahead by making it harder to pass the measures — or to get them on the ballot at all.

The biggest and most immediate fight is in Ohio, where a coalition of abortion rights groups is collecting signatures to place a constitutional amendment on the ballot in November that would prohibit the state from banning abortion before a fetus becomes viable outside the womb, at about 24 weeks of pregnancy. That would essentially establish on the state level what Roe did nationwide for five decades.

Organizers were confident that the measure would reach the simple majority needed for passage, given polls showing that most Ohioans — like most Americans — support legalized abortion and disapprove of overturning Roe.

But Republicans in the state legislature are advancing a ballot amendment of their own that would raise the percentage of votes required to pass future such measures to a 60 percent supermajority. The measure has passed the Ohio Senate and is expected to pass the House this week.

The Republican measure — which would require support from only 50 percent of voters to pass — would go before voters in a special election this August.

“There are a lot of elected officials leading state legislatures that are being unapologetic, brazen, relentless — choose your adjective — about the fact that they don’t care what voters think on this issue and that their ideological stance on this is going to dictate the outcome,” said Kelly Hall, executive director of the Fairness Project, which supports citizen-sponsored ballot initiatives across the country as a check on gerrymandered state legislatures.

Republicans in Ohio have said openly that their efforts to make ballot amendments harder to pass are aimed at blocking abortion rights. They are putting their measure on the ballot in August, typically a time of low turnout. It will not include the word “abortion,” which abortion rights supporters say will make it hard to engage their voters.

The House sponsor of the ballot amendment for the 60 percent threshold argued in a letter to colleagues that without it, “all the work accomplished by multiple Republican majorities will be undone, and we will return to 19,000+ babies being aborted each and every year.”

Mike Gonidakis, the longtime president of Ohio Right to Life, an anti-abortion group, said that he had worked on behalf of the legislative leadership to get 60 House Republicans to publicly declare that they would support putting the amendment on the ballot in August if the speaker brought it for a vote. Mr. Gonidakis spent his family spring vacation in Florida rounding up that support. “This has been a labor of love,” he said.

He sees abortion as a “policy decision,” not a right, and said that policy should be left to the legislature alone. “Our Constitution is for our constitutional rights, not weed or gambling or abortion,” he said.

Republican-led legislatures in five other states are leading similar efforts to block citizen-led measures. The North Dakota legislature this month approved a bill boosting the signature requirement for proposed constitutional amendments and requiring them to win approval in both primary and general elections.

And in Arkansas, after voters last fall soundly rejected a constitutional amendment proposed by the legislature stiffening the requirements to get a measure on the ballot, the legislature simply passed new requirements as state law. Gov. Sarah Huckabee Sanders signed the law last month.

While some legislators have targeted citizen-led initiatives on redistricting, voting rights for felons and legalized marijuana, abortion opponents and supporters alike agree that the Supreme Court’s decision overturning Roe has supercharged the push for citizen ballot measures and Republican efforts to deter them.

Republicans were surprised by how forcefully voters turned out to reject anti-abortion laws last year, even in red states.

In Kansas, the Republican-controlled legislature put forward a ballot initiative that would have reversed a 2019 state supreme court ruling finding a right to abortion in the state Constitution. It was placed on the ballot in the August primary, when turnout is typically low, but abortion rights groups mobilized to defeat it.

In November, voters defeated a similar measure in Kentucky, along with an anti-abortion law in Montana. At the same time, they approved measures to recognize a constitutional right to abortion in Vermont, California and Michigan.

The decision to raise the threshold to 60 percent in Ohio probably was not an arbitrary choice, said Chris Melody Fields Figueredo, executive director of the Ballot Initiative Strategy Center, which works to support progressive ballot measures.

In other red and purple states — Michigan, Kentucky and Kansas — the vote for abortion rights was between 52 and 59 percent.

“When they’re raising the passage threshold to 60 or 65 percent, it’s often just a percent or a couple of points above what has been needed to pass initiatives in the past,” she said.

One Republican lawmaker who opposes the new limits on initiatives in Arkansas, State Senator Bryan King, said he believes the lure of power, not partisan politics, is the driving force behind them.

“I don’t think this is a party issue. This is a control issue,” he said. “It’s trying to fence off challenges to whatever decisions a government makes.” That desire for control has been constant, he said, regardless of which party ruled the state over the past two decades.

Mr. King has joined a lawsuit seeking to strike down the new restraints on ballot initiatives in Arkansas. “One of the core beliefs I was taught in being a Republican is that we should make it easier for citizens to get things on the ballot and challenge what government does,” he said. The new Arkansas law, he said, “simply crossed the line.”

In Missouri, the Republican-led legislature is on the verge of putting a constitutional amendment on the November ballot raising the approval threshold for proposed constitutional amendments to 66.7 percent, from 50 percent. Voters, however, would be unlikely to know that the measure would do that. The proposal specifies that it be described on the ballot only as a measure to require voters to be properly registered U.S. citizens and Missouri residents — which the state Constitution already requires.

The chief sponsor of the measure, State Representative Mike Henderson, did not respond to phone and email requests for comment. In debate on the House floor, Republicans said they were not trying to deceive voters.

Legislatures began accelerating bans and other restrictions on abortion beginning a decade ago, after Republicans took control of more statehouses. Ohio has been at the forefront of those attempts. It was among the first states to attempt a so-called heartbeat law, banning abortion after roughly six weeks of pregnancy, when many women do not realize they are pregnant. (That law passed in 2019 and went into effect after Roe was overturned but has been temporarily blocked by a state court.)

The state made national headlines in July after a 10-year-old rape victim had to travel to Indiana to get an abortion because a doctor said her pregnancy was beyond six weeks.

Republicans in Ohio first filed the measure to increase the percentage of votes required to pass citizen-led amendments a week after the elections in November. It failed to pass, after demonstrators flooded the Statehouse and shouted from the legislative galleries.

Sponsors refiled the measure in the new term, adding the provision for an August election, which is estimated will cost $20 million. Their amendment would also add new requirements to get proposed amendments on the ballot: Proponents would have to collect signatures from at least 5 percent of the residents in all 88 counties in the state, up from the current 44. The measure would also do away with the so-called curing period, which allows the proponents a week to collect additional signatures to make up for those that authorities disqualified.

The State Senate has passed the measure, and the House is expected to approve it this week.

Abortion rights groups say they are trying to gather 700,000 signatures, well above what they need to get their measure on the ballot before the July 5 deadline. And they are finding strong support as they canvass parks, shopping centers, concerts and athletic contests.

“I have circulated for ballot initiatives before. This is the first time in my life that I have not had to explain what I am carrying,” said Cole Wojdacz, the field manager for Pro Choice Ohio and one of the lead organizers for Ohioans for Reproductive Freedom. “People are chasing me down asking me if I have petitions. It’s like an awakening.”

Their first hurdle, however, is the Republican-led initiative in August. They fear it will be hard to motivate voters to the polls on what seems like an esoteric change to ballot law.

Just four months ago, Republicans in the legislature led passage of a law doing away with most August elections because they cost taxpayers too much and, as Secretary of State Frank LaRose argued at the time, had “embarrassingly low turnout.”

Mr. LaRose, a Republican who supports the August election to raise the threshold for ballot measures, added that August elections tend to mean “just a handful of voters” make big decisions. “The side that wins is often the one that has a vested interest in the passage of the issue up for consideration,” he said in his written testimony at the time. “This isn’t how democracy is supposed to work.”

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