The first major abortion case since Chief Justice John G. Roberts Jr. joined the court last month comes before the justices next week, and even though it could be anticlimactic in the end, both sides in the debate view it as a warm-up for even more consequential cases ahead.

The case is Ayotte v. Planned Parenthood, No. 04-1144. If the court divides 5 to 4 with Justice Sandra Day O'Connor in the majority but cannot produce a decision before O'Connor is replaced, then the court will have to redo the case in its next term.

But if not, "it could be a vehicle to revolutionize abortion law if they want to use it," said David J. Garrow, a Supreme Court historian and specialist on abortion law at Britain's Cambridge University.

The case stems from a 2003 New Hampshire law requiring teenagers to tell a parent before getting an abortion. While the law has an exception for girls who would die without the procedure, New Hampshire lawmakers omitted an exception for other non-life-threatening health problems because they felt it would render the law meaningless.

The new law has never been enforced, because two federal courts have said the lack of a health exception made it unconstitutional.

The Bush administration supports the New Hampshire law, telling the court in a friend-of-the-court brief that the case "may have direct relevance" to its defense of the federal law banning the late-term procedure that its opponents refer to as "partial-birth abortion" -- a law that has been struck down by lower federal courts in rulings that the administration has asked the Supreme Court to overturn.

For their part, a coalition of pro-women's-rights organizations, including the National Organization for Women, the Communications Workers of America and the Ms. Foundation, has filed a brief in which it suggests that a ruling in favor of the New Hampshire law could "cast women's constitutional right to choose and the interests it serves into a continuous state of insecurity." The brief is one of 34 friend-of-the-court briefs in the case representing the views of clergy, interest groups and politicians on both sides.

At the heart of the matter is a relatively arcane issue having to do not with whether the law is unconstitutional but with how the court goes about deciding it is unconstitutional.

In 1987, in a non-abortion case, the court ruled that a law could only be struck down "on its face" -- that is, before it goes into effect -- if there is no possible constitutional way to enforce it.

But in a landmark 1992 abortion ruling, Planned Parenthood v. Casey, the court seemed to apply a different standard to state abortion regulations, noting that such rules would run afoul of the Constitution if they posed an "undue burden" to women in "a large fraction of cases."

If the court were to apply the more restrictive 1987 standard to abortion laws such as New Hampshire's, or, eventually, to the federal late-term abortion ban, it would become much harder for abortion-rights advocates to defeat regulations in court. "The substantive potential of it is huge," Garrow said.

Parental involvement laws regarding abortion are politically popular. All but six states have some form of statute that says girls under 18 must involve at least one parent or guardian in the decision to terminate a pregnancy.

As required by a 1990 Supreme Court decision, those laws generally include a "judicial bypass": Teens may avoid telling a parent if they can convince a judge that they would face abuse or that they are mature enough to make the decision on their own.

New Hampshire's version would make it a crime for a doctor to perform an abortion on a minor unless the doctor has written proof that at least one parent has been notified or unless the doctor certifies the girl would die without the procedure.

But unlike the parental involvement laws in most states, the New Hampshire statute does not explicitly let a doctor proceed when, in the doctor's judgment, the girl might be about to suffer serious health consequences short of death.

Opponents of the law say that is unconstitutional because the Supreme Court has said in past cases that state regulations without health exceptions put an "undue burden" on the right to abortion. It is unrealistic, they say, to expect teens facing such pregnancy-related conditions as severe uterine bleeding or sudden spikes in blood pressure to wait for a parent or a court to let them have an abortion.

In the words of a brief filed by the American Civil Liberties Union, which is representing the law's opponents: "The central issue in this case is simple but profound: Can the state omit a medical emergency exception from an otherwise permissible abortion regulation, even when it is undisputed that compliance with the regulation in emergencies will result in serious medical harm?"

New Hampshire counters that such rare crises could be dealt with through the law's existing exceptions. Under the 1987 Supreme Court case, the state argues, the risks of a severe health emergency are too small to warrant invalidating the whole statute.

A brief submitted by New Hampshire legislators who back the law notes that, out of tens of thousands of abortions done on teens in six other states for which records exist going back to 1991, only 11 were "emergency health abortions."

The law's opponents do not dispute that the number of people at risk may be small, but they argue that the risk even to some people, combined with the legal uncertainty the law imposes on doctors, makes the law unconstitutional.

The justices gather for an official portrait. Front, from left: Antonin Scalia, John Paul Stevens, Chief Justice John G. Roberts Jr., Sandra Day O'Connor and Anthony M. Kennedy. Back, from left: Ruth Bader Ginsburg, David H. Souter, Clarence Thomas and Stephen G. Breyer.