If you and the mother are in agreement, and can represent to the family court judge that it is in your child's best interests that you have sole physical and even sole legal custody, certainly a court will enter an order consistent with your agreement. However, termination of parental rights is something entirely different, as it will legally severe the child-parent relationship. If that is what you are talking about, the answer is "no," a parent cannot voluntarily relinquish parental rights under the circumstances you have described. As pointed out by Mr. Carter, if your spouse wishes to file a petition for step-parent adoption and the birth mother is willing to voluntarily relinquish parental rights, then yes, that can be accomplished. I hope this information is helpful to you.
This comment is designed for general information only, and should not be construed to be formal legal advice nor the formation of a lawyer/client relationship.
The bio mom can consent to you having sole legal and physical custody, and if you marry and your wife wants to adopt your son, then bio mom can agree to terminate her parental rights in an action for stepparent adoption. Otherwise, she cannot terminate her rights. Only the state can act to do this, except in very rare situations, such as when a parent is convicted of sexually molesting their own child. I have done a lot of research in this area, and the rationale behind the refusal to allow parents to voluntarily terminate their rights is that it deprives the child of his right to support. Sadly, many a parent would volunteer to terminate their parental rights if it meant getting them off the hook on child support.
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I recommend meeting with a few lawyers to see who is the best fit for your situation. The advice you have received from the lawyers on the site is sound.