The vote comes as the number of measles cases nationwide this year has passed 600.
The measure now heads to Gov. Jay Inslee, who has expressed support for limiting exemptions. The state has seen 74 cases of measles this year. Most of those cases were centered in one county and involved children 10 or younger who were not immunized.
The Centers for Disease Control and Prevention has reported that as of the end of last week, 626 cases of measles have been confirmed in the U.S. so far this year, up from 555 as of a week ago. While 22 states have reported cases, most of the nation's cases are centered in New York City and in nearby Rockland County north of the city.
Democratic state Rep. Monica Stonier of Vancouver said the measure will "reduce the risk that our communities face when an outbreak is possible."
"It keeps kids in school," she said. "It keeps people with compromised immunities safe in their communities."
Washington is among 17 states that allow some type of non-medical vaccine exemption for personal or philosophical beliefs. In addition, medical and religious exemptions exist for attendance at the state's public or private schools or licensed day-care centers. Medical and religious exemptions remain in place under the measure.
Unless an exemption is claimed, children are required to be vaccinated against or show proof of acquired immunity for nearly a dozen diseases - including polio, whooping cough and measles - before they can attend school or go to child care centers.
The state Department of Health said that 4% of Washington K-12 students have non-medical vaccine exemptions. Of those, 3.7% of the exemptions are personal, and the rest are religious.
While the Senate had first sought a bill that would have removed the philosophical exemption for all required childhood vaccines, both chambers ultimately agreed to move forward with the House bill that focused only on the measles, mumps, rubella vaccine - also known as MMR.
Republican Rep. Joe Schmick said the measure was "the wrong direction that we should be moving."
"The parents should be making this call, and they should be the one to decide," he said.
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