The Times notes that they are more common in wider society than is widely believed:
The dangers of marriage between first cousins are to be highlighted by a leading professor, with a warning that their children are at risk of genetic defects.Baroness Deech, a family law professor and crossbencher, will call next week for a “vigorous” public campaign to deter the practice, which is prevalent in Muslim and immigrant communities and on the rise.
However this misses the point, the danger with a one off cousin marriage is not actually that severe, however when they occur repeatedly over many generations all sorts of defects appear, it goes without saying that the communities that practice it in the UK did so long before they came here. Repeated cousin marriages can lead to children being born who are more inbred than the offspring of brothers and sisters, a famous example of this was Charles II of Spain, whose Hapsburg lineage was so entangled that he was born mentally and physically incapable and arguably led to Spain's fall from great power status. His family tree looked like this:
Famous first-cousin marriages include that of Charles Darwin who wed his first cousin Emma Wedgwood, of the china manufacturing family. They had ten children with a poor record of survival and health: three died in childhood or at birth and five were ill or disabled.
He was not exceptional: Albert Einstein married his cousin, Elsa Lowenthal; Queen Victoria married her first cousin, Prince Albert. Of their nine children, one had haemophilia and died at 31; two more were carriers and passed it on to their children.
The level of inbreeding rose with each generation and that is where the problem lies.
The Times article also makes basic factual errors, as Mark points out here.