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Dear Ms. Gentry,
Thank you for posting your request on History Hub!
Ordinarily, the Federal Government does not create or maintain birth, death, marriage or divorce records. Such records are kept by State and/or local governments rather than the National Archives. Birth, death and marriage records are not available for use in the State of Illinois by the public. Please see Illinois Department of Public Health for more information.
We searched Ancestry but were unable to locate a birth record for Joel Sylvester Hampton. However, we suggest that you search FamilySearch Online Birth,Marriage, Death Records for research tips or to ascertain if a birth record for Joel Sylvester Hampton can be located.
We hope this is helpful. Best of luck with your family research!