The Unified Radio and Plasma Wave (URAP) experiment on the interplanetary spacecraft Ulysses is able to detect the Saturnian kilometric radiation (SKR) thanks to the high sensitivity of the receiver and in spite of the remoteness of the planet (8--13 AU). Our knowledge about Saturn comes essentially from the observations by the two Voyager spacecraft. Ulysses allows us to reassess the main properties of the SKR as they had been observed by Voyager 1 and 2: average flux density, spectrum, periodicity, and polarization. A striking difference between the results obtained from Voyager and those obtained from Ulysses is the periodicity of the radio emission linked to the planetary rotation (10 hours 39 min 24 s): The period deduced from Ulysses' observations is not constant and may differ by 1% from that of Voyager. The northern and southern sources of SKR are distributed along magnetic field lines which are fixed in local time. We interpret the source location by a Kelvin-Helmholtz instability developing on the flanks of Saturn's magnetopause, in the morningside. This instability could be at the origin of the acceleration of the particles responsible for the radio emission. Because of the solar wind fluctuations, the position of the magnetopause and the magnetohydrodynamic flow around it are modified so that the zone of Kelvin-Helmholtz instability slowly moves, implying a drift of the radio sources in local time. We examine and discuss the possibility for this drift to be the cause of the variations observed for the SKR period. ¿ 2000 American Geophysical Union