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No, if you give up your visitation rights at this time or even forever -- you do not give up your legal rights to your child unless you agree to sever parental rights -- and that is an entirely different document that you would have to execute and file with the court. FYI, even if you agree to give up visitation or you want to sever your parental rights the courts will generally not let you off the hook from paying child support payments unless there is someone who is willing to adopt the child in your shoes and take over the care and support of the child in your place (such as if the custodial parent remarries and the step parent wants to adopt the child).
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