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The only thing you can do is be honest with the judge and hope for the best. You can not prevent him from having some type of visitation or custody of the child. The courts prefer to grant joint custody to both parents unless there is a reason why this would not be in the best interest of the child.
Since you are hoping to avoid him getting any custody you will want to hire a lawyer to represent you in this matter and fight for no custody but visitation only.