American man marries British woman how do they get him a British passport?

The British spouse would have to apply for the U.S. spouse/partner to have a residence permit ("leave to remain in the U.K.") and then they would have to take up residence in the U.K. for 3 years continuously. (Actually, for 9 months for each of those years). Then the non- E.E.A. partner can apply for naturalization. This is similar to a non-U.S. person obtaining a Green card and the couple living stateside. Perhaps the non-British partner might qualify for a British passport by descent, say if a parent or grand-parent was born in U.K. or a Colony or former Colony, etc. Children of U.S. servicemen and diplomatic personnel born in U.K. do not qualify. A claim to nationality of most of the European Union and European Economic Area countries can also be used to establish work and residence status in the U.K.. which also open up the possibility of naturalization if so desired after the same time period. Check out an official Foreign and Commonwealth Office (FCO) website for the British position.