We measured the rotation period of Jupiter's inner magnetosphere with precision previously unattainable, using 35 years of observations of the Jovian decametric radiation at the University of Florida Radio Observatory at frequencies between 18 and 22.2 MHz. The new rotation period is the weighted mean of 13 independent 24-year average determinations. Each of these was found by measuring the drift of the histogram of occurrence probability versus System III (1965) central meridian longitude over an interval of approximately 24 years. The measured drift was used to correct the System III (1965) period to obtain the new value. Our weighted mean is 9 hours 55 min 29.6854 s, with a standard deviation of the weighted mean (&sgr;) of 0.0035 s. This new rotation period is 7.4&sgr; shorter than that of the System III (1965), indicating that the latter is in need of revision. Our measurements indicate an upper limit of about 4 ms/year on any possible Jovian rotation period drift. ¿ 1997 American Geophysical Union