paternity testing

sooo there's a paternity test you can do after 9 weeks where you just get the mom and dad's blood and it'll figure out the chance of him being the father.. i hear they're super expensive.. i know for a fact my ex is the dad but since he has a new girlfriend, he's denying it.. i told him that if he wants a paternity test, he's paying for it, but i feel like he wont now.. i want a paternity test done so i can prove to him that he is the father, is there any possible way to use insurance or something else to get it paid for????? will a hospital do it????