Baltimore was a little-known, faraway American city to the residents of Crewkerne, a sleepy country town in the west of England, where some of the cottages still have thatched roofs and the big event of each day is going to the pub after work for a drink.

That was before early Saturday morning, when the town's antique dealer was shot to death by an unknown assailant on a Baltimore street while he was chasing the man who had snatched his girlfriend's purse.

Now, the account of the death of Phillip Rouse, 34, is one of the biggest stories in Great Britain. It led the British Broadcasting Corporation news Sunday night. It made the front page of the popular tabloids, including the Daily Express, the Daily Mirror and the Daily Mail.

"Murder in Cold Blood," screamed the big, black headlines of the Express. "Girl Tells How Mugger Shot Briton In Heart."

"It's caused quite a sensation," David Seymour, a BBC news editor in the west British county of Somerset where Crewkerne, population 3,799, is located, said yesterday. "This is one of England's most beautiful counties. It's a caricature of a picture postcard of England. Incidents of shooting here are very, very rare indeed. They happen once every three or four years. You can imagine why people are so shocked."

"I think you're bloody crazy over there," said Malcolm Frith, a BBC reporter who covered the story. "You seem to go around shooting everybody, including the president . . . You could describe the reaction of the people of Crewkerne as being absolutely stunned."

Rouse, who was divorced, was the director of T.R.G. Lawrence, an antique exporting firm, and was considered an expert on antiques. He was on a month-long vacation in America with his 9-year-old son, Karl, and his girlfriend, Anne Bullivant, 21, a barmaid at a Crewkerne pub, whom he planned to marry. It was Bullivant's first trip to the U.S., and her mother, Bridget Bullivant, said yesterday, "She was looking forward to it immensely. I would have liked for her to enjoy her holiday. Obviously, that was not meant to be."

They had been in Baltimore two weeks, visiting Rouse's boyhood friend from Crewkerne, Nigel Lawrence, who lives in the city's Bolton Hill, a white-collar neighborhood of expensive, renovated town houses. It was Rouse's third summer in Baltimore. "He loved it," Lawrence, 34, said yesterday. "They went to the new aquarium. We went to a baseball game. They went to the Inner Harbor. He's a vegetarian, and you couldn't stop him from eating fish -- crabs and clams and lobsters; he was in his glory."

But Lawrence said his 5-foot 10-inch, 150-pound friend had had trouble in Baltimore before. "Phillip was mugged last year. A young guy ripped his pocket off with his money in it. Phillip said to him, 'Why don't we sit down and talk about this and why don't you explain why you did what you just done?' "

The mugger wasn't interested in talking. "He just ripped Phillip off some more," Lawrence said. "That time he just got a punch in the mouth and a few dollars missing."

The night Rouse died, he, Bullivant and Lawrence had gone to Girard's, a disco where every Saturday night is "Discomania." They were walking home shortly before 2 a.m., a distance of about eight blocks that Lawrence had walked many times without incident. "I blame myself a little bit," he said. "We were talking about how safe places were to walk. I've walked that way so many times I consider it safe."

A young man on a black 10-speed bicycle -- Lawrence said he looked to be about 15 or 16 -- rode by, turned around and pedaled past again, this time grabbing Bullivant's purse, which held her wallet as well as Rouse's. Rouse and Lawrence gave chase, Lawrence running fast behind the bike. "I was so close to that bike," Lawrence said. "That's all I was looking at, really. He was looking at me, too. He was scared because I was so close . . . . Yes, I think I shouted something at him. I might have said, 'You, punk' . . . . I didn't see Phillip at all . . . . I heard a bang. It didn't register as a gunshot."

Rouse, according to Baltimore police, hadn't gotten more than a few steps when three men tripped him and wrestled him to the ground at the 900 block of Mason Street behind the Fifth Regiment Armory, headquarters for the National Guard. One man fired a handgun at Rouse's chest, and the three then fled across the parking lot of the State Office Building, police said. The incident occurred at 1:57 a.m. A Baltimore Fire-Rescue ambulance crew pronounced Rouse dead at 2:01.

"Annie stood on the pavement and watched all of it," Lawrence said. "There was no talking, no fighting, no struggling. It was just cold-blooded murder. They couldn't shoot him in the leg, couldn't kick him in the teeth. They had to kill him . . . . If you could've seen the look on his face afterward. God almighty, he was frightened."

Police had no suspects yesterday. Lawrence, who is also in the antiques business and has lived in Baltimore for three years, says he has already raised $5,000, part of a reward fund that he hopes will help find his friend's killer.

Bullivant and Rouse's son returned to England yesterday. Rouse's father, Charles Rouse, a Crewkerne farm machinery merchant, has telephoned the British Embassy in Washington for help in making plans to fly his son's body home.

"This simply wouldn't happen in Britain," Cyril Gray, embassy information secretary, said yesterday. "We don't have violent crime -- aside perhaps from an occasional bank robbery. Basically, the only reason you can have a gun in Britain is for sporting clubs. Of course, we wouldn't dream of telling you how to do things . . . ."

In England, the press called Rouse a hero. The headline of the Western Daily Press: "West Hero Shot By Mugger." The story began: "Friends last night mourned the tragic bravery of West antiques dealer Mr. Phillip Rouse, 34, shot dead by muggers in the American city of Baltimore. He died on the pavement of a city center street only a short distance from his girlfriend."

Elaine Clifton, who works the switchboard at the BBC office in Bristol, was a friend of Rouse and Bullivant and had met them many times at King's Arms, the Crewkerne pub where Bullivant worked. Clifton, who has never been to the U.S., said: "This makes me think the rumors about the violence in America are true . . . . Anne is very quiet. I can't think of a worse person to cope with all of this."

Before leaving for England yesterday, Bullivant told the Daily Express of London the dramatic story of her friend's death:

"It was a beautiful night and we had been having a delicious time . . . . It was the first time we were able to find a babysitter for Karl. So we went out to a few bars and then to a disco.

"As we strolled home, a black kid rode by on a bicycle. He was wearing a jacket with a hood over his head. He passed us, turned, came back behind us and seized my purse, containing about 18 pounds $34 .

"Phillip didn't hesitate. He ran straight after the thief. He was only 20 yards away when three other black men tripped him. One of them had a handgun. He pointed down at Phillip and fired. I screamed and ran over to him . . . . It was all over in a few minutes."

The British press called Baltimore police repeatedly, seeking information on what was the city's 149th homicide of the year. In all of 1979 -- the latest year for which figures are available -- there were 629 homicides (including murder, manslaughter and other unlawful killing) in all of Great Britain.

"They asked me if this was unusual," police spokesman Dennis Hill said. "I told them it was. The whole thing is bizarre. Were the three men and the purse-snatcher working together? Or was it just two separate incidents -- a purse-snatching and then a homicide?"

And in England, the mother of Anne Bullivant said sadly that perhaps there was a lesson to be learned in the death of Phillip Rouse. "This should be an example to British tourists not to give chase. If Phillip had stood still, perhaps he'd still be alive."