Mo. Rev. Stat. § 182.040
After the establishment of a free county library district the legislative body of any incorporated city, town or village in the county which was excluded from the county library district because of the maintenance of a tax supported municipal library established and maintained pursuant to other provisions of this chapter, after approval of the proposed change by the trustees of the free county library district, may become a part of the free county library district by notifying the county commission that the municipality desires to become a part of the free county library district at the beginning of the next fiscal year; and thereafter the municipality shall be liable for taxes levied for free county library purposes at the same rate as is levied for the free county library district in the county.
§ 182.040, RSMo