There is no outright ban on late-stage abortions at the federal level, although most states do not permit the practice.
In 1973, the Supreme Court ruled in Roe v. Wade that Americans have a constitutional right to abortion, but that states could prohibit the practice at the point of "fetal viability" — when a fetus can survive outside the womb, typically at 24 to 28 weeks — with exceptions required for the life and health of the pregnant person.
Since then, the only federal abortion restriction that has been enacted and upheld is a partial-abortion ban, which makes illegal the termination of a fetus already partially outside the womb.
Forty-three states prohibit abortions after a certain point in pregnancy, except when the pregnancy threatens the individual's life or health. A few states also make exceptions for pregnancies resulting from rape or incest.