Yes, as long as the baby was born in US, she/he is a US citizen automatically. To get a US passport, the mom needs to go to the post office where passport applications are accepted with a birthcertificate of the baby. If the mom is trying to go back to her country immediately, there may be a problem because getting a US passport takes a long time now. Here is the detailed info on getting US passport.
http://www.travel.state.gov/passport/get/processing/processing_1740.ht ml
The baby actually has two citizenships now as he/she is also a citizen of the mom's country. Not all countries allow their citizens to have dual citizenship. US let you have two, but some countries, like my native country, Japan, don't. So if the baby's mom is from the country where they don't allow dual citizenship, the baby will have to decide which one to keep when he/she gets older.


