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You can also order a copy of a Missouri birth certificate from the VitalChek website, which is run by Lexis-Nexis, or by calling them to place an order over the phone. They even offer overnight shipping.
However, VitalChek adds on a substantial shipping and processing fee to all vital records orders, not just the overnighted ones. And after placing your order, you may still need to fax them your "tangible interest document" or notarized statement. They are a third party commercial vendor, not a government agency.