the baby's original birth certificate
Key word is original
. A state-certified copy (with official seal) will also be OK, yet the original one from your state is the best. A hospital certificate will not do. Read more here: Required Cruise Travel Documentation & Information | ID, Passport ID & General Documentation | Norwegian Cruise Line
When you return, I strongly suggest that your get a passport for the baby. It saves a lot of hassles and also provides peace of mind.