DNA test prior to birth and mother moving out of state

I'm in a similar situation. A bad news ex-girlfriend turns up pregnant at the end of a short relationship, when it was already questionable if she had been with other guys. She claims it to be mine, then wants nothing to do with me. I'd rather not have to be tied to this woman in any way, however if the child IS mine, I want to take as much responsibility as I can on all levels. She made clear her desire to be away from me as soon as she found she was pregnant. However now at 5 months pregnant she is planning to move to another state.

What are my rights? I have already made clear to the mother my desire to have a DNA paternity test prior to the child's birth. She made clear that would not happen until after the child was born. Personally this tells me she is probably not sure of the father either, however what can I do about requiring and gaining a paternity test prior her moving out of state? Could that keep her from moving? What are my options?

From my non-professional web research, both DNA paternity tests available prior to birth only risk miscarriage by less than 3%. Still a concern I know, but is that enough to keep a court from requiring one in this type of situation?
By the way I live in Arkansas, the mother is planning to move from AR to California.
Thanks,
~Grand

You're not going to get a court to order a DNA test before the child is born, and until you establish your parental rights you're not going to get a court order keeping the child in the state.