We describe our high-resolution measurements of the 133Cs 6p (2)P(3/2) state hyperfine structure. An optically narrowed diode laser excites perpendicularly a highly collimated atomic beam. The spectra are calibrated with a stable reference diode laser using a rf locking scheme allowing us to determine the splittings with an accuracy of < or =2 kHz, an order of magnitude better than previous results. The magnetic dipole a, electric quadrupole b, and magnetic octupole c hyperfine coupling constants are determined. The values we obtained are a=50.288 27(23) MHz, b=-0.4934(17) MHz, and c=0.56(7) kHz. This work represents the first observation of the magnetic octupole moment of the cesium nucleus. We carry out atomic-structure calculations and determine the nuclear electric quadrupole moment Q= -3.55(4) mb and nuclear magnetic octupole moment Omega=0.82(10) b x mu(N).