JACKSON, Miss. — If a girl younger than 16 gives birth and won’t name the father, a new Mississippi law — likely the first of its kind in the country — says authorities must collect umbilical-cord blood and run DNA tests to prove paternity as a step toward prosecuting statutory-rape cases.

Supporters say the law is intended to chip away at Mississippi’s teen-pregnancy rate, which has long been one of the highest in the nation. Critics say that though the procedure is painless, it invades the medical privacy of the mother, father and baby. Questions abound: At roughly $1,000 a pop, who will pay for the DNA tests in the country’s poorest state? Even after test results arrive, can prosecutors compel a potential father to submit his DNA and possibly implicate himself in a crime? How long will the state keep the DNA on file?

Republican Gov. Phil Bryant said the DNA tests could lead to prosecution of men who have sex with underage girls.

“It is to stop children from being raped,” said Bryant, who started his career as a deputy sheriff in the 1970s. “One of the things that go on in this state that’s always haunted me when I was a law-enforcement officer is seeing the 14- and 15-year-old girl that is raped by the neighbor next door and down the street.”

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Bear Atwood, legal director for the American Civil Liberties Union of Mississippi, said it’s an invasion of privacy to collect cord blood without consent of the mother, father and baby. She also said an underage girl who doesn’t want to reveal the identity of her baby’s father might skip prenatal care: “Will she decide not to have the baby in a hospital where she can have a safe, happy, healthy delivery?”

The law took effect July 1, but hasn’t been used yet. Cord-blood samples would have to be taken immediately after birth, and the state medical examiner is setting administrative rules for how the blood will be collected and stored. Megan Comlossy, health-policy associate for the National Conference of State Legislatures, said she thinks Mississippi is the first state to enact a law authorizing the collection of blood from the umbilical cord — a painless procedure — to determine paternity.

Statistics put the state’s teen-pregnancy rate among the highest in the country. In 2011 — the most recent year for which statistics are available — there were 50.2 live births in Mississippi per 1,000 females ages 15-19, according to the Centers for Disease Control and Prevention. The nationwide rate was 31.3.

And more than half Mississippi’s 82 counties reported at least one pregnancy by a 10- to 14-year-old girl in 2011, according to an Associated Press analysis of state statistics.

The bill’s main sponsor, Republican state Rep. Andy Gipson, said the U.S. Supreme Court has ruled that DNA left on objects, such as saliva on a disposable cup, can be tested as evidence in a criminal case. He said he thinks umbilical-cord blood fits that description.

Gipson said he doesn’t believe a man who fathers a child with an underage girl should have a reasonable expectation of privacy. “Most cases would involve a suspect who is pretty well-identified,” he said.

The new law says it’s reasonable to think a sex crime has been committed against a minor if the baby’s mother won’t identify the father; if she lists him as unknown, older than 21 or dead; or if the identified father disputes paternity. The law says health-care workers and facilities cannot face civil or criminal penalties for collecting cord blood, and failure to collect is a misdemeanor offense. The law doesn’t address whether the mother can refuse blood collection or what would happen if she does.