Dictionary.com appears to have 'British' as a noun meaning 'the people who are native inhabitants of Great Britain' however, this doesn't sound correct to me. 'A British' really doesn't work to my ear.
For me, British is only an adjective. I guess a dictionary would be correct, and so it should also be a noun, but I'd disagree. Also, a Northern Irish person is British, but they aren't a native inhabitant of Great Britain, so the dictionary isn't pedantic enough for me.
'Briton' is definately correct.