If an American citizen living in the United Kingdom (England if it matters) meets all of the UK's requirements for naturalization, can he become a British citizen without losing his American citizenship?
From the American perspective:
U.S. law does not mention dual nationality1 or require a person to choose one nationality or another. A U.S. citizen may naturalize in a foreign state without any risk to his or her U.S. citizenship. Source: U.S. Department of State — Bureau of Consular Affairs
And from the British:
Dual citizenship (also known as dual nationality) is allowed in the UK. This means you can be a British citizen and also a citizen of other countries. Source: Gov.Uk
1Section 101(a)(22) of the Immigration and Nationality Act (INA) states that “the term ‘national of the United States’ means (A) a citizen of the United States, or (B) a person who, though not a citizen of the United States, owes permanent allegiance to the United States.” Therefore, U.S. citizens are also U.S. nationals. Non-citizen nationality status refers only individuals who were born either in American Samoa or on Swains Island to parents who are not citizens of the United States [Source: Dept of State, as above]