Asteroid Ida's satellite Dactyl was observed over 5$\frac{1}{2}$ hours by the Galileo spacecraft imaging system. The observed motion fits a family of orbits parameterized by the mass of Ida (Belton \al 1996). We have tested the stability of these orbits by numerically integrating motion about a realistically shaped model for Ida. Those with pericenter distance $q \approxless 65$ km (corresponding to Ida's density $\approxgreater 3.1 \gcm3$) are unstable over time scales of a few days to a few months, placing a strong upper limit on Ida's density. Moreover, at the opposite extreme of density, orbits corresponding to densities less than $2.3 \gcm3$ are chaotic and become unstable after about 1000 yr. For density between 2.3 and $2.5 \gcm3$, Galileo family orbits are chaotic but there is no indication of instability over thousands of years. Dactyl likely formed at the same time as Ida, so its orbit must be stable over timescales much longer than we have been able to explore numerically. As a start toward understanding long-term stability, we have investigated the character of orbits commensurate with the rotation of Ida within the Galileo family. We found that the overlap of high-order resonances for low densities of Ida explain the chaotic behavior of orbits. The low-order $p$:1 and $p$:2 resonances, corresponding to a high density for Ida, are distinct and stable and are all consistent with the longitudinal position of Dactyl at the epoch of the Galileo encounter. However, there is no evidence of preferential stability of resonant orbits against collision with Ida or escape over 6000 yr. If a resonant orbit is actually occupied, it may have been selected by a longer-term stability or by dissipative processes.