This is already an ugly case. 607 of the IMDMA prohibits visitation with a sex offender (or in the presence of) unless he's taken a rehabilitative class. Since both you and your ex have serious sex offender issues, I suspect neither of you want this issue in front of a judge anytime soon.
My advice, contact your local legal aid for assistance before someone opens the Pandora's box.
It is not against the law for you to date anyone. It is also not against the law for you to live with anyone.
That said, if you do move in with your current boyfriend, there will be legal consequences. First, you will need to provide your ex husband notice of your residency with a sex offender under Chapter Section 609.5 of the Illinois Marriage and Dissolution of Marriage Act (the "IMDMA") which provides as follows: "Sec. 609.5. Notification of remarriage or residency with a sex offender. A parent who intends to marry or reside with a sex offender, and knows or should know that the person with whom he or she intends to marry or reside is a sex offender, shall provide reasonable notice to the other parent with whom he or she has a minor child prior to the marriage or the commencement of the residency."
Further, your living with your current boyfriend may be a basis for you ex husband to seek to modify the current custody situation under Section 610 of the IMDMA, subsection (a-5) which provides: "A motion to modify a custody judgment may be made at any time by a party who has been informed of the existence of facts requiring notice to be given under Section 609.5 [750 ILCS 5/609.5]." The court will then determine custody based upon the facts of your case, with the best interest of the child in mind. The court will have a difficult time with your case since your ex husband is also a registered sex offender. Therefore, there is likely to be a hearing over custody and how to best arrange for custody of the child. This does not mean that your ex husband will necessarily get custody of your child, but it does mean that there is likely to be court proceedings should you move in with your boyfriend.
As an aside, under 607(e) of the IMDMA, no parent or stepparent of any minor child, convicted of any offense involving an illegal sex act perpetrated upon a victim less than 18 years of age . . . is entitled to visitation rights while incarcerated or while on parole, probation, conditional discharge, periodic imprisonment, or mandatory supervised release for that offense, and upon discharge from incarceration for a misdemeanor offense or upon discharge from parole, probation, conditional discharge, periodic imprisonment, or mandatory supervised release for a felony offense, visitation shall be denied until the person successfully completes a treatment program approved by the court. Therefore, your ex husband might not be entitled to visitation depending upon the circumstances.
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Do you have a divorce decree that in any way restricts your dating? If not you are free to date whomever you like, but this looks like a rebound thing with the same kind of guy, or worse. Believe me, he didn't know she was 12 - 13? Give me a break. What you have going for you is the fact that it is not cheap to go to court and modify a court decree that may have awarded you custody of your daughter; on the other hand in more rural areas the judges can get really picky about good moral character and even if your x is or was a registered offender he or she will look at you, not him, as the parent to be judged for behavior of potentially exposing your CHILD to someone who can't tell ages apart...... Your situation is a tragedy waiting to happen. Legally, there's no "law" but you are risking a LOT.