Dactyl was discovered in solid state imaging (SSI) data on February 17, 1994, during the long playback of approach images from the Galileo spacecraft's encounter with the asteroid 243 Ida. Forty-seven images of the Ida-Dactyl pair were obtained. A detailed search for other satellites was made. No confirmed detections were made, all other candidate features being consistent with radiation hits. We deduce a manifold of osculating two-body orbits that approximate Dactyl's motion over the observed orbital arc depending on the assumed mass of Ida. At the time of Galileo's encounter, Dactyl was found to be 85 km from the center of Ida, moving at ∼6 m·sec-1in the same rection as Ida's retrograde spin. The inclination of its orbit is ∼172° in Ida's equatorial system (IAU definition). It was not possible to obtain a definitive orbit or measure of Ida's mass from the observed motion even though supplemental techniques (search for Dactyl's shadow on Ida, changes in angular diameter and brightness, and attempts to determine the spin of Dactyl) were explored. The influence of Ida's irregular gravitational field and solar perturbations on two-body motion are evaluated and found to be undetectable in the observed orbital arc. These effects may, however, strongly influence the motion over orbital time scales. Limits to the value of Ida's gravitation parameter,GM, are derived. A robust lower limit,GM> 0.0023 km3·sec-2, is obtained by requiring Dactyl's orbit to be bound. Hubble Space Telescope observations, which show no evidence of Dactyl on a hyperbolic orbit, excludes values ofGMin the range 0.00216 <GM< 0.0023 km3·sec-2. An upper limit,GM< 0.0031 km3·sec-2, deduced by requiring that the orbital motion has a long lifetime in a realistic approximation to Ida's gravitational field, is illustrated with numerical calculations. Ida's mass is therefore constrained to the range 4.2 ± 0.6 × 1019g, which, together with a volume of 16,100 ± 1900 km3(Thomas P. C., M. J. S. Belton, B. Carcich, C. R. Chapman, M. E. Davies, R. Sullivan, and J. Veverka 1996.Icarus120, 20-32.) yields a bulk density of 2.6 ± 0.5 g·cm-3(Belton, M. J. S., C. R. Chapman, P. C. Thomas, M. E. Davies, R. Greenberg, K. Klaasen, D. Byrnes, L. D'Amario, S. Synnott, T. V. Johnson, A. McEwen, W. Merline, D. R. Davis, J-M. Petit, A. Storrs, J. Veverka, and B. Zellner 1995.Nature374, 785-788.).