A bill easing access to late-term abortion in Massachusetts passed in the state House of Representatives on Wednesday. The Senate is expected to vote on the measure before the legislative session ends next week.

If the bill becomes law, Massachusetts would be among the states with the fewest restrictions on abortions late in pregnancy.

Current state law limits abortion after 24 weeks of pregnancy, with several exceptions that advocates have said are not enough to ensure that all women in need of abortion care can get it without leaving the state. Those exceptions allow late-term abortions if the life of the pregnant woman is in danger, if her physical or mental health is at risk, if there is a “lethal fetal anomaly” or if there is a “grave fetal diagnosis.”

The new bill would allow an abortion after 24 weeks if a doctor deemed it appropriate.

“We found that this care is so complicated, there is no way to encompass every possible exception,” said Representative Christine Barber, a Democrat who sponsored the legislation. “It should be between a doctor and a patient.”

Massachusetts has a Democratic governor and a Democrat-controlled legislature.

The House approved the bill in a 119-33 vote. For it to become law, the Senate must approve it before the current legislative session ends on July 31. A spokeswoman for Gov. Maura Healey did not immediately respond when asked on Wednesday if she would sign it into law.

Massachusetts Citizens for Life, an anti-abortion group, said in a statement that while the bill’s sponsors had “good intentions,” the bill raised serious concerns.