Mo. Rev. Stat. § 550.140

The clerk of the court in which any criminal cause shall have been determined or continued generally shall, immediately after the adjournment of the court and before the next succeeding term, calculate all costs which have accrued in the case. If the state or county is liable under the provisions of this chapter for costs or any part thereof, he shall make out and deliver forthwith to the prosecuting attorney of said county a complete fee bill, specifying each item of services and the fee therefor. The circuit clerk shall prepare a bill against the state containing only costs which are payable to the county.