We describe the first in situ search for Jovian X radiation with a dedicated X ray instrument. Upper limits to Jupiter's X ray power in the 27--48 keV range are presented using data taken by the gamma ray burst experiment on the inbound Ulysses trajectory, at distances between 163 and 110 RJ . The data were analyzed using three methods for detection of (1) a steady flux, (2) a flux linked to System III rotation, and (3) a sporadic flux. The 3&sgr; upper limits lie between 1.1¿108 and 5.6¿108 W. The results are consistent with the idea that the bulk of the Jovian UV auroral emissions are due to energetic electrons, while the previously observed low-energy X radiation is due to ion precipitation. ¿ American Geophysical Union 1993