I have this same problem. My son is 19 also, girlfriend 20. Now her mother is trying to convence her not to put him on the birth certicate, and she is thinking about moving to Georgia we all currently live in Arkansas. So what are his rights, and ours as grandparents? Can we get a court order ordering her to stay in Arkansas or to take a DNA test? She is almost 9 weeks, there is one paternity test that can be done between 10-16 weeks. Can we force it with an order or does she get to decide if she wants to do it?

Your son has no rights until the child is born when a DNA test is done.

There certainly is a test that can determine DNA between 10 - 16 weeks (amniocentesis), but I don't know any doctor who will do that strictly for determination of paternity. There are risks such as miscarriage and rupture of membranes associated with this kind of test.

Bottom line is to contact a Family Law attorney. The likelihood is that you cannot do anything until the child is born. There is a slim chance that the mother can be enjoined from moving until a paternity test is taken, but a local attorney will know better the mood of the courts in such matters.

If she runs away and hides, there may be nothing that can be done. If she stays or moves but keeps in contact, then your son will need to file for custody as soon as the child is born. Where will depend on where the child is born. Once a legal action for custody is entered, a paternity test will be required.

One last point, I would document all your son 's attempts to step up and be a father if the child is his. If the mother disappears and then, later tries to get child support, this documentation may help prevent being hit with a whopping back support bill.

The problem herein lies that the OPs son is only presumed to be the father until a DNA test is conducted. I don't know of any physician who will put their license on the line to conduct an amniocentesis strictly for the purpose of paternity.

Right, and a court is unlikely to act on the presumption. They will not order a pre-natal test. The most they might, and even that is very much a long shot, is require that the mother stay in the jurisdiction to have the baby or keep the potential father aware of her location.

The problem herein lies that the OPs son is only presumed to be the father until a DNA test is conducted. I don't know of any physician who will put their license on the line to conduct an amniocentesis strictly for the purpose of paternity.

The problem herein lies that the OP's son is only PUTATIVE father until a DNA test is conducted.
There's legal difference between presumed and putative father.