Both Chief Justice John Roberts and Justice Samuel Alito are considered conservatives.
Justice Alito is thought to be more likely to rule against abortion than his predecessor, Justice Sandra Day O'Connor.
The supporters of the South Dakota law say they want to trigger a battle over the 1973 Roe-versus-Wade ruling, in which the US Supreme Court established that governments lacked the power to prohibit abortions.
'Vulnerable and helpless'
About 800 abortions are performed each year in South Dakota. Planned Parenthood, which performs abortions in the state, has pledged to challenge the measure in court.
Pro- and anti-abortionists recently marked 33 years of Roe v Wade
Under the law signed by South Dakota Governor Mike Rounds, doctors could get up to five years in prison for performing an illegal abortion.
Gov Rounds said: "In the history of the world, the true test of a civilisation is how well people treat the most vulnerable and most helpless in their society.
"The sponsors and supporters of this bill believe that abortion is wrong because unborn children are the most vulnerable and most helpless persons in our society. I agree with them," he said in a written statement.
Court costs
The abortion ban would take effect on 1 July but it is likely that a federal judge would suspend it during any legal challenge.
The law would therefore not take effect unless South Dakota state gets the case to the US Supreme Court and wins.
Gov Rounds said abortion opponents have already started offering money to help the state pay legal bills for the anticipated court challenge.
Lawmakers also said an anonymous donor has pledged $1m (£572,000) to defend the ban, and a special account has been set up to accept donations for legal fees.
The statement issued by Gov Rounds also noted the bill was written to ensure existing restrictions on abortion would be enforced during a legal battle.
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