Family Law Questions? Ask a Family Lawyer Online.
There is no legal mechanism where a parent can voluntarily give up her parental rights and responsibilities. In order to do this, one of two things generally has to happen. Either the Department of Child Services (DCS) has to move to terminate your rights for parental unfitness- they want to take the child and put it foster care or elsewhere; or there must be a petition to adopt the child, by either a step-parent or someone else.
In your case, the more likely scenario would be if there was someone else who wanted to adopt the child and step into the mother role. You cannot voluntarily leave the child with only one parent, so there would have to be someone else wanting to adopt in order for you to give up your rights. The courts favor the child having access to two parents, even though you have no contact with the child, there are other legal rights like inheriting.
Sorry, I know it is not what you wanted to hear, but this is the reality of your situation. My policy is to just give the truth, rather than tell you what you want to hear now, only for it to end in more frustration for you later.