[For part I see authors, ibid. 26, No. 4, 807-821 (1993; Zbl 0772.60092) and for part II see K. Briggs and authors, ibid. 27, 1503 (1994).] The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.