Recently my ex girlfriends daughter has shown interest in living with me, me becoming her father.

If I get a DNA test land it is shown that her daughter is mine, can I legally adopt her?

If it is negative, what rights would I have?

What age is the child? If you are the father you will have to prove beyond a doubt that the child's mother is no fit to take care of her. Then you might be able to go for custody. If you are not her father I don't believe you have any rights concerning the child. On the other hand, I do believe if the mother feels she can't care for her child and wishes for someone else to do so, you may both agree that the child lives with you on a temporary basis with the eventual custody agreement being drawn up. If the mother is not in agreement with any of this I don't think you will be able to get the child.

If you do get DNA and you are the father you don't need to adopt her. You will just need to file for custody/visitation orders.

If you aren't, then what Bluerose says would more than likely be the case.