The law also says doctors who perform such abortions could face fourth degree felony charges and lost their medical license.
"This is not an issue about abortion; it is an issue of discrimination - discriminating against a person, not allowing them their God-given right to life, simply because they might have Down syndrome", CNN reported Rep. Sarah LaTourette, the legislator who introduced the bill, said earlier this month.
In the case of Roe v. Wade in 1973, the Supreme Court created a right to abortion even though none existed in the Constitution.
Down syndrome is a genetic condition that affects cognitive ability, causing mild to severe learning disabilities and distinctive facial characteristics. But abortion rights groups argue the law will be another blow to women's constitutional right to legal abortion.
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Kasich signed Ohio House Bill 214 Friday, roughly a week after the legislation was passed by the state assembly. According to a 2012 study originally published in Prenatal Diagnosis, families and individuals choose abortion between 50 and 85 percent of the time following a fetal diagnosis of Down syndrome.
Opponents say such a law would damage doctor-patient relationships and does nothing to improve lives of those with Down syndrome.
In November, after Ohio's House and Senate passed the bill, the American Civil Liberties Union of OH countered, saying they oppose the bill as it sets a precedent for politicians to interfere in the lives of women. A similar law in North Dakota went into effect in 2013 and has not been challenged.
The law would come into effect in 90 days.
North Dakota and IN have passed similar restrictions. The Republican governor had previously vetoed the "heartbeat bill", which would have banned abortions after about six weeks of pregnancy, but signed another bill prohibiting abortions after the fifth month, or 20 weeks.