Muhammad was found guilty of killing Mr Meyers in a petrol station
The US Supreme Court has rejected an appeal for a stay of execution from lawyers for the man behind the 2002 sniper attacks in Washington DC.
John Allen Muhammad is due to be executed on Tuesday for the murder of Dean Harold Meyers, one of 10 people killed during three weeks of attacks.
The court gave no reason for the rejection of the clemency appeal.
Muhammad, 48, has one final appeal for clemency pending before the Virginia Governor, Timothy Kaine.
Muhammad's then teenage accomplice, Lee Boyd Malvo, is serving life in prison for his part in the attacks.
The two men killed six people in Alabama and Louisiana before moving on to Maryland, Washington and Virginia.
Mr Meyers was shot at a petrol station in Manassas, Virginia.
Muhammad will die by lethal injection at a prison in Virginia but his lawyers have criticised the court ruling.
"Virginia will execute a severely mentally ill man who also suffered from Gulf War Syndrome," the Associated Press news agency quoted lawyer Jonathan Sheldon as saying.
Mr Sheldon also criticised the timing of the execution, due the day before Veterans Day, a US public holiday honouring the military.
Correspondents say the execution revives memories of a time when the Washington area was paralysed with fear.
7 years on Washington residents recall sense of fear while sniper was on the loose
People were shot while shopping, filling up their car at a petrol station or just sitting and reading, always with a single round and from a distance.
After three weeks, Muhammad was arrested at a truck stop, along with Lee Malvo, who was only 17 at the time.
While they killed 10 people around Washington, the pair are also suspected of murders elsewhere.
Lawyers say John Muhammad has never shown remorse.
Cheryll Witz is one of several victims' relatives who are going to watch his execution. Her father, Jerry Taylor, was shot by Malvo - at Muhammad's direction - on an Arizona golf course in March 2002.
"He basically watched my dad breathe his last breath," she told the Associated Press.
"Why shouldn't I watch his last breath?"
HOW MUHAMMAD AND MALVO MURDERED THEIR VICTIMS
Killers targeted shopping malls, schools and petrol stations
Modified blue Chevrolet Caprice used as cover and firing point
Pair used high-power sniper rifle and scope
Victims were killed by single shot fired from distance