STOCKHOLM, Sweden--A man shot Prime Minister Olof Palme down on a snowy sidewalk yesterday night as he walked with his wife in downtown Stockholm. He was pronounced dead at a hospital soon afterward.

Police were hunting "a darkhaired man aged 35-40 in a long dark overcoat" who apparently fled in a car.

Palme, 59, and his wife, Lisbeth, had attended a movie premiere and were at an intersection when the man opened fire with a handgun, witnesses said. Police said he was hit at close range by at least two bullets in the chest and stomach.

Bystanders and Mrs. Palme tried to help the prime minister as he lay bleeding in the snow, and a taxi driver called for an ambulance and police. He was taken to Sabbatsberg Hospital, his wife at his side in the ambulance, and died on the operating table shortly after midnight, hospital sources said.

Police closed off the entire capital to search for the killer. A police report said squad cars chased a Volkswagen sedan headed north.

"We have very little to go after so far, with only sketchy witness reports," police superintendent Sune Sandstrom said.

Ingvar Carlsstrom, the deputy prime minister, arrived at Government House at 1 a.m. today to lead a crisis meeting of the Cabinet. The domestic news agency, TT, quoted him as saying: "We have asked all those we could get to come over here. It is horrible. I just got to know it 30 minutes ago."

Palme led the Social Democrats, Sweden's socialist party. He first was prime minister from 1969-76, then returned to office in 1982 after leading his party to victory over an incumbent conservative coalition. He won re-election to a three-year term in 1985.

He first became known outside Sweden in the 1960s, as an outspoken critic of the U.S. role in Vietnam.

While he was education minister in 1968, he marched side-by-side with a North Vietnamese diplomat at an anti-American rally in Stockholm, which inspired complaints from Washington and demands by the Swedish political opposition that he resign.

He was an aristocrat who chose socialism, and a brilliant debater with a sharp tongue that frequently got him into trouble.

He once said about his change in political loyalties: "I am born in the upper class but I belong to the labor movement. I have come to join the labor movement by working for the working class on its own conditions and by adhering to a movement which desires liberty, equality and fraternity between people."

In 1984, Palme gave the first Wurf Memorial Lecture at the Kennedy School of Government. Speaking during the process of instituting some economic reforms in Sweeden, Palme examined the development of modern socialism in his north European country.

Palme's visit to Harvard later generated controversy. Some of his critics in Sweden charged that the Prime Minister had, with the help of Harvard officials, rejected a highly taxable $5000 stipend he could have earned from his Harvard speech to finance tax-free his son's study here the following year.

Palme denied those charges, but the Sweedish Press treated the Harvard-related incident as a sizable scandal during the latest election in that country.

He was born into a bourgeoise Stockholm family with noble ancestors on January 30, 1927. He has three sons.

As a student he traveled to many countries, including the United States. He said seeing poverty and social misery at close range awakened him to socialism.

After private schooling with top grades, Palme went to the United States and earned a Bachelor of Arts degree at Kenyon College, Ohio, in 1948.

He was a secretary on the defense staff when Prime Minister Tage Erlander noticed him and opened the road to politics. He was named secretary to the government in 1954 and gained ministerial rank in 1963, the youngest man to do so in Europe at the time.

Palme served as minister for communications 1965-67 and led the Education Ministry from 1967-1969. He succeeded Erlander as leader of the Social Democrat Party in 1969 and began his first term as prime minister in October.