MPs were under pressure to overturn the ruling
The bill, passed with the support of the government, follows three cases in which judges ruled that families whose children were born with birth defects could sue because doctors did not spot the problems during pre-natal scans.
The rulings outraged French disabled groups and doctors. Specialists who carry out the scans have been on strike for over a week.
Passed by the National Assembly, the bill now goes before the upper house of parliament, the Senate.
The government hopes to have the legislation in the statute books by the end of February, thus bringing an end to a year-long moral and legal controversy before parliament is suspended prior to June elections.
In November 2000, France's highest court awarded damages to Nicolas Perruche, a teenage boy born with severe mental and physical disabilities.
His mother contracted rubella during the pregnancy and she argued that if doctors had correctly diagnosed the illness, she would have had an abortion.
In effect, the boy sued the doctors for having been born.
Ruling 'devalues lives'
Two recent similar rulings involving children with Down's syndrome have re-ignited anger within France's disabled community, which sees the decisions as devaluing the lives of those born with disabilities.
"Look at my son and tell me he should have been aborted," said Martha Lebihan, standing next to her 15 year-old son, Morgan, who is autistic, as she addressed reporters on Wednesday.
"He is a wonderful and extraordinary child who brings us so much happiness," she said.
It has also led to growing concerns within the medical community.
Doctors say that because no ultrasound scan can be 100% accurate, they are under increasing pressure to advise abortions, even if there is only minimal concern.
Prenatal specialists have seen their insurance premiums soar to up to 10 times their previous rates since the Perruche ruling.
'Blatant error'
Ethicists and legal specialists have also attacked the rulings.
"To allow a child to be born cannot be considered as a mistake - that must be written into law," said Laurent Aynes, a professor in civil law at Paris' Sorbonne University.
The government had held out against legislation, but was forced to act by public pressure and by the decision by some medical staff to stop carrying out prenatal scans.
The bill passed on Thursday says no-one can claim damages for having been born.
Parents - but not the children in question - will still be able seek damages, but only on the grounds of a "blatant error" by doctors.