San Mateo County has tentatively approved what is believed to be the first measure in the nation to require owners of dogs and cats to either buy a breeding license or get their pets sterilized.

Over the protests of pet owners, the Board of Supervisors approved the measure 4-1 Tuesday after a four-hour hearing that drew breeders from across the country. The ordinance faces a second vote next week before becoming final.The ban on unlicensed breeding is intended to reduce the number of unwanted animals put to death.

Under the ordinance, effective in 1992, professionals and amateurs would have to obtain a license to breed their pets or else spay or neuter them. Rules for the program, such as the cost of the license, have not been worked out.

The ordinance would affect only unincorporated areas of San Mateo County, or about 10 percent of the 628,000 people in the primarily suburban, well-to-do county immediately south of San Francisco. Thus, it would not apply to such communities as Redwood City, San Mateo, Menlo Park and South San Francisco.

"We just lost," said Betty-Anne Stenmark, a member of Responsible Dog Breeders of San Mateo County.

She and others said they may challenge the ordinance in court.

The Peninsula Humane Society has said some 10,000 stray dogs and cats a year are put to death in San Mateo County. Breeders argued that they were not responsible for the pet overpopulation problem. They blamed that on stray animals.

During the meeting, some in the audience hissed rock singer Grace Slick of the Jefferson Starship as the animal rights activist spoke in favor of the ordinance.

"We're users, the human race. One of the things we use the most are the animals," she said.

Outside, one of about 50 demonstrators against the ordinance carried a sign reading, "If the Humane Society had been in Hollywood, there would be no Lassie."