I am waiting for DNA test results of a newborn the mother claims I fathered. I have no issues with this. But what are my rights as to how often I can see this child and the length of time the newborn is with me and my current family?

That will be decided by the court. Up to 50% (or more if you fight hard enough or the mother agrees). When paternity is established, you file for joint physical custody and fight out in court how much time you get. Or you get visits - every other weekend sort of thing.

Really, its up to you, the mother and the judge.

I agree with everything in the above post but also know that:

You can fight in court, but it can be an expensive process.
Make sure that once paternity is established you try your hardest to work with the mother on a schedule that you can both be comfortable with.
If you can't agree, then fight through court.
If you can agree, it's the best possible scenerio for you.

First, please pay more attention to posting guidelines. The Children forum (where this was originally posted) has a Read First sticky that direct legal questions here.

Was the DNA test ordered by a court? Is the mother filing for support through the courts? Does she have an attorney?