The answer you get depends on how you ask the question. There is no just "signing over" parental rights, unless what you are talking about is adopting the child. In that case, she would sign a Consent to the adoption, and her parental rights would be terminated by the court as a part of the adoption process.
If you are talking about a change of the custody status alone, her parental rights remain, but there is no longer a right to have input into decisions.
This response does not create an attorney-client relationship and is intended for general information purposes only.
You have 2 options. The first is to enter into an agreement whereby the mother gives up joint legal custody and all visitation except the one weekend per month and her child support obligation is terminated. However, this is not the same as terminating her parental rights and can be modified in the future. The second option is for the mother to consent to you adopting the child through a step-parent adoption.
This answer is provided for general information purposes only and does not constitute an attorney-client relationship. To receive advice tailored to your unique circumstances, you should meet with an experienced attorney.