Dual citizenship

Joe is looking for further clarification on the dual citizenship issue raised in the June 27 Lifelines column.

"I am a U.S. citizen," he says, "married to Japanese. We have no kids as yet. My job has me traveling extensively, and my wife would prefer to give birth in a Japanese hospital due to language concerns. I don't know about the eligibility for U.S. citizenship in this case either (I was born abroad to two American parents, but what if the parents are different nationalities?).

I had heard that dual citizenship applied in this case, but that was most likely rumor. Could you please help me with the research, and inform us all about this 'what if' scenario?