Russell Bucklew, 46, was scheduled to die at 00:01 local time on Wednesday.
His lawyers say a congenital deformity in his throat would put him at risk of choking during a lethal injection.
The execution would be the first since a botched lethal injection last month.
Oklahoma prisoner Clayton Lockett died of a heart attack in April after the executioners reported trouble finding a vein in which to inject the lethal drugs and officials halted the execution. He reportedly writhed and moaned as the first drug of three lethal drugs, a sedative, apparently failed to take effect as intended.
Russell Bucklew’s reprieve, however temporary, came after a day of legal wrangling that first saw a three-judge panel of the 8th Circuit Court of Appeals, a regional federal court, grant him a stay of execution, only to have that stay overturned hours later by the full court.
Then, with just two hours to go until the scheduled execution, Supreme Court Justice Samuel Alito, who handles emergency appeals for the 8th Circuit region, issued his own stay.
Russell Bucklew could be put to death later on Wednesday if the Supreme Court considers his case and opts not to intervene.
He suffers from a condition called cavernous hemangioma, which causes malformation of the blood vessels and tumors in the throat and nose.
In an affidavit for his defense team, an anaesthesiologist and surgeon wrote last week that during an execution Russell Bucklew would be “at great risk of choking and suffocating because of his partially obstructed airway”.
In 1996, Russell Bucklew murdered a man he believed was romantically involved with his ex-girlfriend, then raped the woman during his escape from the scene of the murder. He was convicted of first-degree murder, kidnapping, burglary, forcible rape, and armed criminal action.
The US has had increasing trouble in recent years finding drugs to use in executions, amid an embargo from the EU, where pharmaceutical manufacturers have refused to ship drugs to buyers in the US for use in lethal injection.
The shortage has led the states in some cases to turn to lesser-regulated compounding pharmacies.
Missouri executes inmates with a single, lethal dose of the drug pentobarbital.
Like other states, Missouri keeps the source of the drug secret, in an effort to shield the provider from scrutiny and unfavorable publicity. News organizations and inmates have sued the states in an effort to force them to disclose the companies names. Their efforts have been rejected by the courts.
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