Custody would automatically pass to the surviving parent. You will have to file for a modification of support if you are already exercising more custody. Once the mother passes, a death certificate should be forwarded to the support office at which point they will proceed to terminate the support.
I agree with Mrs. Townsend who has provided an excellent answer. Your rights as the father supersede those of her family (i.E. child's grandparents) so you would be entitled to custody.
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I also agree with Mrs. Townsend.
I'll add -- don't rely on a phone call. Go the court now and file a motion to terminate support. Mom can designate someone else as "guardian" in her will, but you can't "will" a child - your rights will trump and unless there would be grounds to terminate them in a DYFS case, you get custody "by operation of law." It's not a "best interests" test.
Anyway - go to court and do this in writing. Don't assume they'll call back. In Burlington, it's 49 Rancocas Road in Mt Holly.
The above is said without seeing your case file and without my understanding the entirety of the facts of your case. Depending on those facts, the above information be may incomplete or may be completely inaccurate. The above is intended as general information only based on what you described and not as legal advice. I advise you to consult with counsel who may be able to provide better information commensurate with a better understanding of your situation.
Excellent answers by previous contributors. You wiil need to take active steps, such as filing a motion with the Court, not just phone calls. Remember the poeple you are calling dont work for you ,,they are not your advisers either. Even, before the mother passes away, you can get it done, if you are able to show, that she is unable to take care of the child becasue of her terminal illness. As a biologaical parent, your right can not be secondary, unless it is established that you are unfit parent.