There is a great deal of information missing from your question for your question to be properly answered. I will try to answer. There is a presumption that if a woman is married that her husband is the parent of any children born to that woman. This presumption is enhanced if the husband is on the birth certificate. That having been said, if the woman seeks child support from your husband and it is established through a paternity test that he is the father, he may be ordered to pay child support.
If your husband believes he is the father of the children and he wants to have visitation and/or custodial rights, he can petition for such rights. The court would order a paternity test and if it were establised that he is the father, he most likely would be awarded some visitation and/or custodial rights. Additionally, if the mother were to file an action for child support, your husband would be entitled to counter sue for visitation and/or custody.
I hope that this answers your question. I suggest that you speak with an attorney to obtain a more complete answer.
Please be advised that any information or advice given herein does not constitute an attorney client relationship. Further, before taking any actions or deciding not to take any actions, you should seek counsel of an attorney. This forum cannot provide you with the anyalysis and detail necessary for a full and accurate opinion to be rendered.
Basically, the only sure-fire way to permanently extinguish a current and/or future obligation to pay support for your minor child(ren) is if the child(ren) are adopted by another person (this also terminates various other rights and obligations, such as inheritance rights, etc.) This is true even if another individual has court-ordered residential custody of the child(ren), or de facto residential custody. When you state that the new husband has "custody" I assume you mean that the children reside with him full-time, and that your current husband has no court order providing him visitation with the children. This will not extinguish his potential liability to pay child support. That said, the mother of the children must approach requesting child support from him with caution, as the court will almost certainly allow for reasonable access and visitation with the minor children for the biological father (assuming he requests it). Occasionally, this possibility deters the mom from filing for a variety of reasons, such as she may not want the children to know that their step-father isn't their biological father, there may be too much bad blood between the parents and court ordered visitation would be too stressful on the mom, etc. The most important thing to remember about any type of family law question is that the number one thing the court is concerned about is what is in the best interests of the child(ren). This determination trumps almost everything else, and (assuming the court properly words it), gives a judge or a master tremendous discretion to formulate a court order. I hope this helps! If my answer created more questions than it answered, feel free to email me for clarification.
This advice does not/is not intended to establish an attorney-client relationship. This advice is general in nature, and is naturally limited in scope and applicability to the limited factual scenario presented in your question. This advice should not be relied upon by you and you should take no action as a result of this advice unless and until you have met with the advising attorney and apprised her of all pertinent facts associated with your inquiry. You should immediately consult with a licensed attorney for advice specific to your case. Moreover, as reasonable minds often disagree, you may find that advice rendered to you based upon your circumstances may vary based upon the opinion(s)/experience and other intangible factor(s) comprising the experience and knowlege of the advising attorney. You are, therefore, encouraged to seek another opinion. Reliance on the advice given in this answer is entirely at your own risk, and you specifically agree that there shall be no liability whatsoever or redress for you, of any sort, for any damages or other cause(s) of action against the advising attorney.