We report measurements of the hyperfine coupling constant for the $8p \
^2P_{1/2}levelofatomiccesium,^{133}Cs,witharelativeuncertaintyof\sim0.019%.OurresultisA = 42.933 \: (8)MHz,ingoodagreementwithrecenttheoreticalresults.Wealsoexaminethehyperfinestructureofthe8p
\ ^2P_{3/2}state,andderivenewvaluesforthestateenergiesofthe8p \
^2P_{1/2}and8p \ ^2P_{3/2}$ states of cesium.Comment: 8 pages, 7 figures, 5 table