The high court rejected an appeal from Texas authorities, who argued that the lawyer's inattention did not necessarily equal an unfair trial.
The Supreme Court's action means that Texas must choose whether to retry Calvin Jerold Burdine or set him free.
CBS News Correspondent Barry Bagnato reports this case has become the poster child for death penalty opponents who claim poor defendants often get stuck with bad lawyers.
The decision not to decide seemed fairly basic to CBSNews.com Legal Analyst Andrew Cohen.
"I don't think the Justices, or enough of them anyway, wanted to go on the record as ruling that it's legally OK or constitutionally fair for a capital defendant's lawyer to sleep through significant portions of the trial," he said. "Had the Court intervened in this case that's pretty much what the message would have been."
Burdine was convicted of stabbing to death his gay lover, W.T. Wise, at the Houston trailer they shared in 1983. Burdine confessed to police, but now denies killing Wise. He claims an accomplice actually killed Wise, while Burdine tried to talk him out of it.
Jurors and a court clerk later described how court-appointed lawyer Joe Cannon slept for up to 10 minutes at a time during the 1984 trial, and the separate sentencing phase that followed. Burdine was sentenced to death.
Burdine came within moments of execution in 1987 before receiving a court-ordered reprieve.
Cannon, who has since died, denied falling asleep.
Burdine lost several rounds of appeals before a federal court agreed that Cannon's performance violated Burdine's constitutional right to an effective lawyer. A federal appeals court panel first reversed that finding in a highly criticized ruling in 2000. The full appeals court then agreed to hear the case, and agreed with the first court that Burdine did not get a fair chance to defend himself.
Texas' appeal to the Supreme Court claimed that the ruling by the 5th U.S. Circuit Court of Appeals creates a conflict among federal appeals courts about what constitutes unacceptable lawyer conduct.
"This isn't necessarily a signal one way or the other that the Court is evaluating these bad lawyer cases differently," said Cohen. "Last week it upheld a ruling making similar but certainly not identical claims. I just think the facts of the Burdine case were so egregious that the Court couldn't allow the conviction to stand."
The Supreme Court sat on the case for months, apparently because it had agreed to hear a case that raised similar issues about the right of appeal when a death row inmate claims his lawyer was inadequate.
Last week, the court ruled against the Tennessee inmate involved in the other case. The court majority found his lawyer's performance was not bad enough to justify an exception to strict rules intended to shorten death row appeals.
It was not clear Monday why the court acted as it did in Burdine's case. Instead of sending the case back to the appeals court for reconsideration in light of the Tennessee case, the high court rejected Texas' appeal outright.
"I don't think the Court wanted to send it back to the lower appeals court and take the chance that those judges might find a way to wiggle out of giving Burdine a new trial or even freeing him," said Cohen.
The Supreme Court has not yet taken a case that asks the broader question of what to do about under-prepared or overworked death penalty lawyers. Away from the court, two justices have expressed strong reservations about the quality of legal help that some inmates receive.
Harris County District Attorney Chuck Rosenthal said Monday he will get a new trial set for Burdine after he examines the Supreme Court's decision to not take up the state appeal.
The case is Cockrell v. Burdine, 01-495.
The Supreme Court Monday also:
Justices will decide if an appeals court wrongly opened the government to liability from the Navajo Nation - and potentially many other tribal governments in future cases.
At issue is whether a federal agency failed to protect the tribe's interest in mining leases on reservation land and must pay for it.
The contracts allow state utilities to purchase electricity in advance, at a set price. Duke Energy, a power marketer, claimed that California utilities defaulted on power contracts during the state energy crisis, and that federal law allowed suppliers to respond by having the contracts voided. The energy could then be resold at higher prices.
The court turned back a German family's challenge to the U.S. government's taking of the paintings and photographs - along with the family's claim for millions of dollars in damages.
Justices will look at the conduct of a California officer who insisted on interviewing the man in a hospital emergency room while doctors repeatedly asked police to leave their patient alone.