Below is a link to a site w/some info. You can have google translate the page. Search on "stuttgart Geburtsurkunde"
Some highlights from translated page:
" If the exact place of birth in Stuttgart is not clear for you, the Stuttgart registrars clarify."
"US citizens who are born in the US Army Hospital in Bad Cannstatt (5th General Hospital). For Information we have a Step-by-Step Guide to order a Birth Certificate. "
Can’t speak German. I contacted the US Army Garrison in Stuttgart and they referred me to a German website. But I’m having trouble with it. Thank you for your help.