On May 7, 1997, the Galileo orbiter flew through the magnetosphere of Ganymede and crossed flux tubes connected at both ends to the satellite. Energetic electrons, observed during this encounter by means of the Energetic Particle Detector on board Galileo, showed double loss cones and "butterfly" type pitch angle distributions, as has been noted in past publications. In addition, as the spacecraft flew toward Ganymede, both the shape and magnitude of the spectrum changed. The intensities decreased, with the greatest depletion observed at the lowest energies, and the monotonic slope characteristic of the Jovian environment was replaced by a rollover of the spectrum at the low-energy end. The spectra lead us to infer a strongly energy-dependent injection efficiency into the trapping region. As on previous encounters, the pitch angle distributions confirmed the position of the magnetopause as indicated by the magnetometer measurements, but the spectra remained Jovian until the trapping region was reached. Various physical mechanisms capable of generating the observed spectra and pitch angle distributions, including downstream reconnection insertion followed by magnetic gradient drift and absorption of the lowest-energy electrons by Ganymede and injection from Jovian flux tubes upstream are assessed.