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I'm sorry to hear about your situation. Assuming that no "acknowledgement of paternity" was ever signed, he doesn't have rights to the child now, but could still petition the court for a paternity determination and ask for custody / visitation rights. Once genetic testing is completed and paternity is established, he'd then become an adjudicated father. Adjudicated father is defined in Texas Family Code Section 160.102(1) as a man who is determined to be the father of a child by the Court. Therefore, once paternity is established by results of genetic testing then the Court will name him as the father and proceed forward with child support, visitation, etc.
In contrast, Texas Family Code Section 160.204 defines a presumed father as follows:
(a) A man is presumed to be the father of a child if:
(1) he is married to the mother of the child and the child is born during the marriage;
(2) he is married to the mother of the child and the child is born before the 301st day after the date the marriage is terminated by death, annulment, declaration of invalidity, or divorce;
(3) he married the mother of the child before the birth of the child in apparent compliance with law, even if the attempted marriage is or could be declared invalid, and the child is born during the invalid marriage or before the 301st day after the date the marriage is terminated by death, annulment, declaration of invalidity, or divorce;
(4) he married the mother of the child after the birth of the child in apparent compliance with law, regardless of whether the marriage is or could be declared invalid, he voluntarily asserted his paternity of the child, and:
(A) the assertion is in a record filed with the bureau of vital statistics;
(B) he is voluntarily named as the child’s father on the child’s birth certificate; or
(C) he promised in a record to support the child as his own; or
(5) during the first two years of the child’s life, he continuously resided in the household in which the child resided and he represented to others that the child was his own.
(b) A presumption of paternity established under this section may be rebutted only by:
(1) an adjudication under Subchapter G; or
(2) the filing of a valid denial of paternity by a presumed father in conjunction with the filing by another person of a valid acknowledgment of paternity as provided by Section 160.305
The signing of a birth certificate is not "magic" in that it doesn't necessarily make your husband the father, nor does it take away any rights the true biological father has. The biological father is still the biological father, and has rights to seek custody/visitation upon establishment of his paternity. Now that being said, even if it's true that he's the father, the court wouldn't necessarily award him custody then and there. The court makes a determination of custody based upon the "best interests of the child", and if the child has been living with you for 5 years, the court's not going to take the child away from home and put the child with a stranger, unless the court determines that the child is living in a situation in your home that is dangerous to the child's physical or mental well-being. So the biological father does have rights to ask the court to give him custody, but that doesn't mean he has a good chance of getting it. But he could still get visitation if he can establish his paternity through a DNA test.
Hope that clears things up a bit. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Thank you, ***** ***** luck to you!