We report measurements of the hyperfine coupling constant for the 8p 2 P 1/2 level of atomic cesium, 133 Cs, with a relative uncertainty of ∼0.019%. Our result is A = 42.933 (8) MHz, in good agreement with recent theoretical results. We also examine the hyperfine structure of the 8p 2 P 3/2 state, and derive new values for the state energies of the 8p 2 P 1/2 and 8p 2 P 3/2 states of cesium.