Rape victims would have to contact police within 50 days, and doctors would have to confirm those reports with police.
Doctors also would have to take DNA samples from aborted fetuses and give that information to police.
In cases of incest, doctors also would have to get the consent of the woman and report the crime to police with the name of the alleged perpetrator.
The measure also would allow abortions to save women's lives and in cases where their health would be seriously jeopardized unless they have abortions.
Sponsors say the exceptions should meet the objections of voters who rejected last year's proposal to ban most abortions.
That bill contained an exception only when a woman's life was in danger.
The new abortion bill has 25 sponsors. Last year's failed bill had 41 sponsors. HB1293
A bill that would ban most abortions in South Dakota contains exceptions for rape and incest, but doctors could do abortions in those instances only if the crimes were reported to authorities.