No Senate Democrats voted to override Beebe's veto of the 20-week measure last week, and only two of the 48 House Democrats supported the override.
However, the top Democrat in the House said he expected fewer Democrats to sustain Beebe's veto this time around.
"I've heard from some members who've gone back home since the vote on the veto override last week and have felt quite a bit of pressure from their constituents," said Fayetteville Rep. Greg Leding. "We pushed as hard as we could last time and came up short, and I don't expect to see that kind of effort this time."
At least one Democrat who supported Rapert's bill said he was unlikely to override the governor. "I think it's probably less constitutional than the first one was," said Sen. Larry Teague, a Democrat from Nashville who voted for Rapert's bill.
The Senate approved the 12-week ban on a 26-8 vote Thursday, moments after approving the override of the 20-week prohibition along party lines. The House approved it a week earlier on a 68-20 vote.
The measures are among several abortion restrictions Arkansas lawmakers are considering after Republicans won control of the House and Senate in November. Beebe signed one of those measures, prohibiting insurers participating in the online marketplace created under the federal health law from covering most abortions.
The American Civil Liberties Union of Arkansas has called the 12-week measure "the most extreme abortion ban in the country" and has vowed to sue if it's enacted.
"Despite continued attacks on women's health care from the Arkansas Legislature, Governor Beebe has remained steadfast in his support for women's health care," said Jill June, president and CEO of Planned Parenthood of the Heartland. "Unfortunately we can't say the same about extreme lawmakers pushing these dangerous bills."
The original version of Rapert's bill would have banned abortions as early as six weeks into a pregnancy, but he changed the measure after facing resistance from some lawmakers.
The measure was written so that it would take effect 90 days after the Legislature adjourns. The Legislature is not expected to adjourn until later this month or next month.
Andrew DeMillo can be reached at www.twitter.com/ademillo