This paper focuses on the dust environment between the orbits of the Galilean moons of Jupiter. Recent discovery of dust clouds around the Galilean satellites formed by impact ejecta from hypervelocity impacts of interplanetary micrometeoroids <Kr¿ger et al., 1999d> suggests that a fraction of the ejected particles may escape from the source satellites into circum-Jovian orbits. We estimate production rates and study dynamical evolution of the escaping ejecta, controlled by gravitational, radiation pressure, and electromagnetic forces, to show that grains larger than several tenths of a micrometer in radius are likely to stay in bound orbits around Jupiter for tens or hundreds of years until they either are lost to collisions with the satellites or Jupiter or are ejected to interplanetary space. It is concluded that these small debris form a broad dust ring with number densities up to ~103 km-3, extending at least from Europa's orbit outward beyond the orbit of Callisto. Our results are consistent with in situ measurements of the Galileo spacecraft. We analyze impact events recorded by the Galileo dust detector from 1996 through 2001 and find more than 200 events outside the orbit of Europa, compatible with impacts of particles orbiting Jupiter in prograde orbits. An empirical dust number density distribution derived from these data agrees quite well with the theoretical one.