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.