Do you know what year and/or state your grandfather was born in? I found a list of American Begeys from the 1935 census (
http://tinyurl.com/z7myxor).
Did your great-grandparents or grandfather have any adoption documents or his birth certificate? It might take a little while, but you could always lobby the adoption agency (if it still exists), the hospital, or the government for access to them. Knowing if your biology has a history of diseases is a big concern anyway, so you could give that as a reason for wanting to know.