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From the Nebraska State Historical Society web site:

BIRTH AND DEATH RECORDS for Nebraska were not consistently kept prior to 1904 at either the state or county level.

The Historical Society does not have official records of births and deaths on file, with the exception of an OMAHA BIRTH REGISTRY, 1869-1907; an OMAHA DEATH REGISTRY, 1873-1915; and a few vital records for early eastern Nebraska, which can be found in THE NEBRASKA AND MIDWEST GENEALOGICAL RECORD. Some earlier birth and death records exist for Lincoln and Omaha and have been filed with the Bureau of Vital Statistics. For available records of births and deaths after 1904, contact the Nebraska Health and Human Services, Bureau of Vital Statistics, 1033 "O" Street, Suite 130, Lincoln, Nebraska 68508, 402-471-2871 to obtain prices and procedures.