The geneticist said that it was common in the Islamic world for men to marry their nieces and cousins.
He said that Bradford has a particular problem and warned that it could affect the health of children born into these marriages.
Prof Jones, who lectures at University College London, is likely to find himself at the centre of controversy in the wake of the comments.
Similar remarks made by Phil Woolas, a Labour environment minister, in 2008 resulted in calls for him to be sacked from the government.
Prof Jones, who writes for the Telegraph’s science pages, told an audience at the Hay Festival: “There may be some evidence that cousins marrying one another can be harmful.
“It is common in the Islamic world to marry your brother’s daughter, which is actually closer than marrying your cousin.
“We should be concerned about that as there can be a lot of hidden genetic damage. Children are much more likely to get two copies of a damaged gene.”
He added: “Bradford is very inbred. There is a huge amount of cousins marrying each other there.” Research in Bradford has found that babies born to Pakistani women are twice as likely to die in their first year as babies born to white mothers, with genetic problems linked to inbreeding identified as a “significant” cause.
Studies have found that within the city, more than 70 per cent of marriages are between relations, with more than half involving first cousins.
Separate studies have found that while British Pakistanis make up three per cent of all births, they account for one in three British children born with genetic illnesses. Prof Jones also said that incest was more common than is often realised in every part of society, adding that it had been particular prevalent among royalty and suggested it is still -continuing.
“Royal families are the human equivalent of fruit flies because they do all the sexual experiments you can think of and there are some examples of inbreeding.
“Royalty did it to keep the heritage within the family line.
“Inbreeding doesn’t apply particularly to our own royal family, but there is some.”
He explained that Prince Charles and Diana could both be traced back to Edward I, with Prince Charles being able to do this through 3,000 “lines”–overlapping connections between people in his family tree–and his former wife being able to do it through 4,000, making the Princess of Wales “from stronger aristocratic heritage” than her husband.
“Their parents had much ancestry in common,” he said.
“We are all more incestuous than we realise.
“In Northern Ireland lots of people share the same surname which suggests a high level of inbreeding.
“There’s a lot of surname diversity in London but if you look at the Outer Herbrides there are rather fewer surnames in relation to the number of people.”