There is no proceeding to accomplish what you are looking for. There is no way for one parent to divest the parent of all of their rights, forever, unless there is an adoption. If a parent has failed to support a child, it may be possible to have a step-parent adopt the child without the need for consent by the non-supporting but if the parent is disabled by his mental problems and can't support the child, that may be another matter. If there is no order for parenting now, then the father has no rights that are in effect now. He does retain residual parental rights and the time to deal with those is when he seeks contact with the child or your new husband wishes to adopt the child.
THE ANSWER PROVIDED is for general informational purposes only. This attorney does not intend to give legal advice. Evaluating the statutes and case law, as well as any relevant court documents together with a more detailed factual analysis could result in a different response. The attorney does not intend to create an attorney/client relationship by the response. Anyone seeking legal advice should always consult with retained legal counsel for a full evaluation of his or her claims.
As stated in the first answer, there is no way to stop the father from having rights to the child until someone adopts her. However, if the father does have any court ordered rights, you can go to court and file to have all contact between the father and child terminated so that he would have to return to court to re-establish his ability to see the child or otherwise talk to those (teachers, doctors....) who are involved with the child. If there is no court order at this time, then he only has the rights that you allow him to have to be involved with the child.