There may be a record, but by law, adoption records, including the birth cert, are sealed and can only be opened by court order at the request of the adoptee, adoptive parents, or in the case of identifying info, the birth parents -- in this case, all of these are dead, except for the presumed adoptee.
Without a court order by the adoptee to open the record, there's no way, as far as I can tell, to even know if a record exists.
Here's a good site with link to state statutes, and a downloadable pdf file of a summary of state laws:
In many states, even the adoptee doesn't have legal standing to force the records to be opened [which is a hugely controversial topic in American family law].