  We show that the Casimir effect should be influenced by variations of the
gravitational potential. This could be tested with a satellite in a highly
elliptic orbit. Still significant technology development is required to achieve
a relative accuracy of $<5 \times 10^{-10}$ necessary to detect the effect
around Earth. That may be obtained in the future based on recent concepts for
giant casimir forces. Such a space mission could evaluate both the laws of
gravitation and quantum mechanics and their interaction. A dedicated mission,
Gravity-Probe C, is proposed.
