Experts say one in every 100 babies are affected by their mothers drinking while pregnant.
More than 6,000 are born every year with Foetal Alcohol Spectrum Disorder which can cause learning difficulties, hyperactivity, co-ordination problems, brain damage and deafness.
But one in 10 mothers-to-be believe that drinking doesn't do as much damage as experts claim, according to the survey by baby charity Tommy's.
A total of 1,300 women were questioned for the survey.
Prof Andrew Shennan, of Tommy's, said: "These findings are worrying because if women don't want to admit to others they are drinking then they might not admit to themselves how much they are drinking.
"Drinking too much can cause serious neurological as well as behavioural problems for the child."
The survey also found that one in 10 women did not know how much a unit of alcohol is.
Prof Shennan said: "With young women in particular who are used to binge drinking there is a need for better education.
"We don't want to be a nanny state but at the same time women need to be well informed."
Nice, the health watchdog, recommends women should not drink at all in the first three months of pregnancy.
They should then drink no more than two units a week, which equates to two small glasses of wine.