  We report measurements of the hyperfine coupling constant for the $8p \
^2P_{1/2}$ level of atomic cesium, $^{133}$Cs, with a relative uncertainty of
$\sim$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 $8p
\ ^2P_{3/2}$ state, and derive new values for the state energies of the $8p \
^2P_{1/2}$ and $8p \ ^2P_{3/2}$ states of cesium.
