The Oklahoma Legislature approved a bill on Thursday prohibiting abortions after about six weeks of pregnancy, a ban that could sharply reduce abortion access not only for women in the state but for those who have been crossing its borders to work around increasingly strict anti-abortion laws across the South.

The bill is modeled on one that took effect in Texas in September. It bans abortion after cardiac fetal activity, generally around six weeks of pregnancy, and requires enforcement by civilians, allowing them to sue any doctor who performs or induces the abortion, or anyone who “aids or abets” one. The bill incentivizes lawsuits by offering rewards of at least $10,000 for those that are successful.

Gov. Kevin Stitt, a Republican, had signed a law this month that outlaws abortion entirely except to save the life of a pregnant woman “in a medical emergency,” and makes the procedure a felony punishable by up to 10 years in prison and a $100,000 fine.

But while that law will not take effect until late August, the bill the Legislature sent to Mr. Stitt’s desk on Thursday would take effect immediately if signed.

Planned Parenthood said its clinic in Oklahoma had seen a 2,500 percent increase in the number of patients from Texas in the months after that state’s law took effect. Anticipating that the governor will sign the six-week bill, some clinics say they have stopped scheduling patients because they expect to be overwhelmed by women facing the new deadlines.

“It will be chaos for plenty of our patients,” said Emily Wales, the interim president at Planned Parenthood Great Plains. “We already have too few appointments. Every time we get a new schedule it will fill in a matter of hours.”

The clinic had been referring patients to clinics in Colorado and Illinois, she said. But with waiting periods and other restrictions already in effect, “there are absolutely people who cannot figure out the logistics of getting farther.”

Mr. Stitt had laughed about the flood of new patients from Texas when he signed the criminal ban this month, suggesting that the state’s highway patrol could “arrest the Texans that come across our border.”

“This bill will take care of that,” he said.

Mr. Stitt has said “we want Oklahoma to be the most pro-life state in the country.”

Despite the earlier criminal ban signed by the governor, the Republican-controlled Legislature has been considering several other anti-abortion measures, including the one passed on Thursday. The strategy is an attempt to layer bans on top of one another and close off any loopholes.

Oklahoma is also considering a bill that would ban abortion 30 days after the start of a woman’s last monthly period, before many women realize they are pregnant. Because the state has a 72-hour waiting period, that law, too, would effectively ban abortion.

The bill passed on Thursday would allow exceptions to save the life of a mother in a medical emergency and also provides an exception for pregnancies that result from rape or incest, as long as those crimes are reported to law enforcement.

State Senator Julie Daniels, a Republican and a sponsor of the bill, said it would “save many innocent lives by acting as a deterrent to those who perform abortions.”

The new laws fly in the face of Roe v. Wade, the 1973 Supreme Court decision that established a constitutional right to abortion. But they are consistent with a trend in other states because of an expectation that a conservative majority on the court will overturn or significantly scale back the Roe decision by summer.

Roe prohibits states from banning abortion before a fetus is viable outside the womb, which is now about 23 weeks of pregnancy. The court is now considering a Mississippi law that bans abortion after 15 weeks of pregnancy, and in oral arguments in December a majority of justices signaled they would uphold it.

States were encouraged by the Texas law, which the Supreme Court has repeatedly declined to block. The court said that because state officials were not responsible for enforcing the law, it could not be challenged in federal court based on the constitutional protections established by Roe. In March, the Texas Supreme Court similarly declined to block the law, saying its hands had been tied by the requirement for civilian enforcement.

In March, Idaho became the first state to pass a similar ban. But the Idaho Supreme Court stayed the law after Planned Parenthood sued to stop it, arguing that the law undermined privacy.

On Thursday, a coalition of abortion providers in Oklahoma filed lawsuits against the Texas-style bill that was passed, as well as the earlier ban scheduled to take effect in August.

Ms. Wales, the Planned Parenthood president, said the Idaho Supreme Court’s decision had given the coalition some hope it could block the Oklahoma bill.

A spokeswoman for Mr. Stitt declined to comment on Thursday, writing in an email that his office “does not comment on pending legislation.” But the governor has said before that he will sign any anti-abortion legislation that crosses his desk.