WASHINGTON — Gov. Tim Kaine said Tuesday that he would not stay tonight’s scheduled execution of John Allen Muhammad, the man dubbed the “D.C. Sniper” whose murderous shooting spree in the Fall of 2002 left at least 10 dead.

In a written statement, Mr. Kaine said: “I find no compelling reason to set aside the sentence that was recommended by the jury and then imposed and affirmed by the courts. Accordingly, I decline to intervene.”

On Monday, the Supreme Court refused to intervene in the case involving Mr. Muhammad, 48, who was sentenced to die for the killing of Dean H. Meyers, an engineer who was shot in the head at a gas station in Manassas, Va.

Mr. Meyers was one of 10 people killed in Maryland, Virginia and Washington over three weeks in October 2002. Mr. Muhammad’s accomplice, Lee B. Malvo, who was 17 at the time, was sentenced to life in prison without parole. The two are also suspected of fatal shootings in Alabama, Arizona and Louisiana.

The execution will bring to a close a case that has fixated the region ever since area residents were gunned down while doing the most mundane tasks, like shopping or pumping gas. The random nature of the shooting spree left people fearful and led many to remain indoors as much as possible to avoid becoming a target.

When the police announced that witnesses had reported having spotting white box trucks near the sniper shootings, the public became obsessed with the ubiquitous work vehicles and a sense of panic often beset anyone sitting at an intersection near the trucks. After a teenager was shot outside his Maryland school, local officials decided to keep schoolchildren inside at recess and they began drilling on duck-and-cover techniques.

Mr. Muhammad’s execution will also end a hard-fought legal battle.

His current lawyers lodged a last set of emergency appeals with the Supreme Court last week, arguing that Mr. Muhammad suffers from severe mental illness and brain damage, caused partly by childhood beatings. The lawyers have also argued that the case has moved too quickly.

While the Supreme Court did not comment in refusing to hear Mr. Muhammad’s appeal, three justices objected to the relative haste accompanying the execution.

Justice John Paul Stevens complained that “under our normal practice,” Mr. Muhammad’s petition for the court to take his case would have been discussed at the justices’ conference scheduled for Nov. 24. But because Virginia scheduled the execution for Tuesday, the judicial process was rushed, Justice Stevens said in a statement joined by Justices Ruth Bader Ginsburg and Sonia Sotomayor.

Justice Stevens wrote that he did not disagree with the majority’s decision to decline the case. However, in declining to stay the execution, he said, “we have allowed Virginia to truncate our deliberative process on a matter — involving a death row inmate — that demands the most careful attention.”

After Mr. Muhammad was sentenced to death in Virginia for shooting Mr. Meyers, Maryland prosecutors arranged to have him tried again for the six murders in Montgomery County. At that trial, Mr. Malvo, who is now 24, testified at length. Throughout both trials and a number of subsequent appeals, Mr. Muhammad continued to profess his innocence.

A soldier-turned-auto-mechanic, Mr. Muhammad held a deep grudge against his ex-wife and society. During the Maryland trial, Mr. Malvo testified that the aim of their shooting spree was to create havoc to cover for Mr. Muhammad’s plans to kidnap his three children.

The longer-term goal, Mr. Malvo said, was to extort law enforcement to stop the killing, after which Mr. Muhammad would take the money and move to Canada with Mr. Malvo and his three children. There, Mr. Malvo said, Mr. Muhammad planned to create a training ground for 140 young homeless men whom he would send out to wreak similar havoc and to “shut things down” in cities across the United States.

Although Governor Kaine has said in the past that he is personally opposed to the death penalty, he has allowed a number of executions to take place since he took office in 2006. Virginia has the nation’s second-busiest death chamber, behind Texas.

Prison officials said the execution process will begin around 8:30 p.m. at Greensville Correctional Center, a state prison in Jarratt, Virginia. An execution team will strap Mr. Muhammad to a gurney, attach him to a heart monitor and connect an intravenous catheter to each arm.

Prison officials will then open a curtain so witnesses in an adjoining room can watch the proceedings. Shortly after 9 p.m., the executioners will inject Mr. Muhammad with a series of chemicals, ending with a fatal dose of potassium chloride, according to prison officials.

Under Virginia law, a prisoner is allowed to choose the method by which he or she will be put to death — either lethal injection or the electric chair. Because Mr. Muhammad declined to select a method, by law he will receive a lethal injection.