Don't worry right now about what will happen when the child turns three years old. File an application for child support with the Attorney General just as soon as they will let you. That way it will be in the works and your name will be moving up their list. If you'd rather have a private attorney that is perfectly fine--they will be much more responsive to you, and act more quickly. Know you can refuse the AG's services whenever you want. It is nice to have the AG enforce a child support order that your private attorney obtains for you, over the course of years for free. I think you may very well be able to limit visitation to Texas until the child is three. Texas Judges seem to really respect breastfeeding, and it is better for your baby anyway, so do that, and express milk while you are at work if possible. Breastfeeding provides a good reason not to have the baby away from you for too long (and therefore for too far) during the first year. You may also want to ask, when you do go to court to establish paternity, custody, child support and visitation, for a provisions to prevent international abduction, and for you to have control of the child's passport. Texas will not be impressed that he choose not to move near his child. He will not be able to go from not really being around the child to taking the child to another State when the child turns three, as long as you have followed my advice and either went through the AG or a private attorney to get custody orders as soon as possible after the child is born. If he violates the court's possession order it could be considered kidnapping. Enjoy the moment --what you are experiencing is precious.
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As soon as the baby is born you need to get orders for child support and visitation. You can get a geographic restriction on the child's residence - maybe to the state of Texas. You may be able to get a restriction on international travel. Hire a lawyer.
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Where was the child conceived? What are the father's ties to Texas?
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It sounds like Texas will have child custody jurisdiction (the ability to make decisions about who has what conservatorship and possession rights to your child) but it's not clear that Texas will have personal jurisdiction over your child's father (to order child support payments). Regardless, what you'll want to do is to file a suit affecting the parent-child relationship as soon as your child is born, in order to get custody orders put in place - it would probably be wise to have the lawyer retained in advance so that the process can be started as soon as the child is born. The odds that a Texas court will let the father take your child to Virginia, either now or when he or she is three, seem pretty low.