The researchers studied the FADS2 gene involved in processing omega 3 fatty acids found in foods such as salmon, nuts and avocados and turning them into nutrients for the brain.
They found breast-fed children had a higher IQ by about 6 to 7 points, but only if they had a variant that made the gene more efficiently process fatty acids.
For those with the less common -- and less efficient -- variant, breast-feeding made no difference when it came to intelligence, the researchers said.
The variant to the FADS2 gene, involved in processing fatty acids, is found in about 90 percent of people, they added.
"For 100 years, the intelligence quotient has been at the heart of scientific and public debates about nature versus nurture," Terrie Moffitt of Kings College London and colleagues wrote in their report, published on Monday in the Proceedings of the National Academy of Sciences.
"Evidence that nature and nurture work together drives a nail in the coffin of the nature-versus-nurture debate."
The study looked at 3,200 children in Britain and New Zealand.
Breast-feeding has many advantages for children including reducing infections, respiratory illnesses and diarrhoea, earlier research has shown. A study presented at an American Heart Association meeting on Monday added healthier blood cholesterol levels to that list.
Although scientists have been looking at potential links between breast-feeding and intelligence for decades, the direct relationship has not always been clear.
The researchers ruled out alternate explanations, saying the effect of FADS2 applied equally to babies with normal and low birth weight. The effect was also the same no matter the mother's social class or IQ.
The team also tested the mothers' DNA and concluded that the FADS2 gene did not somehow alter the quality of breast milk.
"We took cells from the children and then analyzed DNA and then we compared how they scored on IQ tests and looked up if they were breast-fed as babies," Moffitt said in a telephone interview. "It was very straightforward."
While the researchers have found at least one gene linking intelligence and breast-feeding, they said they still need to better understand how FADS2 processes nutrients in breast milk.
They also acknowledge that many other genes also likely have a role in intelligence but say their study offers a different approach to unraveling the human genome, Moffitt added.