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From The Guardian:

Marriage between first cousins doubles risk of birth defects...

Is this true?

After all, the Rothschilds (richest banking family in human history) are known for inter-marriages between cousins and they are still very rich today. So, their offspring cannot be too bad.

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No, except for some subcommunities.

PLoS Biology published an article reviewing the evidence:

Until recently, good data on which to base an answer were lacking. As a result, great variation existed in the medical advice and screening services offered to consanguineous couple. In an effort at clarification, the National Society of Genetic Counselors (NSGC) convened a group of experts to review existing studies on risks to offspring and issue recommendations for clinical practice. Their report concluded that the risks of a first-cousin union were generally much smaller than assumed—about 1.7%–2% above the background risk for congenital defects and 4.4% for pre-reproductive mortality—and did not warrant any special preconception testing.

However, the quote from the Guardian is talking about a specific community - babies of Pakistani origins, in Britain and Wales.

Both the scientists quoted by the Guardian and the 2008 acknowledge this is a more difficult situation:

There may be higher genetic risk in a sub-population:

In the Pakistani immigrant population, for example, the quoted high average rate of birth defects may mask a single trait (or small number of traits) at very high frequency, a situation with different medical consequences from one characterized by a larger number of less-frequent disorders.

Further, there may be other confounding factors:

Inbred populations, including British Pakistanis, are often poor. The mother may be malnourished to begin with, and families may not seek or have access to good prenatal care, which may be unavailable in their native language.

[...]

UK Pakistanis are less likely to use prenatal testing and to terminate pregnancies [20,25]. Thus the population attributable risk of genetic diseases at birth due to inbreeding may be skewed by prenatal elimination of affected fetuses in non-inbred populations.

Whether the additional risks can be used to justify a ban on cousin marriage is outside the scope of this answer. (Wikipedia's Cousin Marriage page gives more background), but, in any case, using a single example of a very rich family is anecdotal, and is a poor justification for dismissing the concerns.

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    I thought that the problem was less one of isolated cases of cousins marrying, but of a culture where most or all marriages were between cousins, so leading to pedigree collapse and concentrating the genetics to a far greater extent. A notable case was King Charles II of Spain whose Hapsburg jaw was so pronounced that he could not chew. – Henry Nov 3 '13 at 12:42
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    @Oddthinking Doesn't "1.7%–2% above the background risk" mean doubling risk if the background risk is 1.7%–2% (in the context of the source)? Consider what I am saying in my answer. – DavePhD Feb 29 '16 at 20:43
  • @DavePhD: Interesting... – Oddthinking Feb 29 '16 at 21:13
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Yes, according to Fact Sheet 18 | WHEN PARENTS ARE RELATED – CONSANGUINITY:

If parents are unrelated, their risk for having a child with a birth defect or disability is between 2% and 3%. If parents are first cousins, the risk is a little higher at 5% to 6%. This is due to the increased chance that they will both carry the same autosomal recessive mutation, passed down through the family.

For more-primary research see the studies in table III of Genetic Counseling and Screening of Consanguineous Couples and Their Offspring: Recommendations of the National Society of Genetic Counselors Journal of Genetic Counseling April 2002, Volume 11, pages 97-119.

The first study Jaber et al. (1998) researched 9 populations spanning the USA, Norway and the Middle East and found 2.1% general risk of "major malformations" versus 4.5% risk for first cousins.

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