Paternity Tests?

Quick question, is anyone else having to get a paternity test for legal reasons? My SO and I have been together for a long time and we obviously know our daughter is his, but because we don't want to get married, we have to have a proven DNA test for things like insurance and so on. 
Will the hospital do it after birth or do we have to go to a testing facility? I'd like this to be as easy and quick as possible so when we go home from the hospital we can get her put on his insurance and stuff.