The speed in the rest frame of the medium is always the same (for a given frequency). It does not depend on the speed of the source (even though the frequency does due to the Doppler effect). The point that Fizeau was making with his experiment is to show that the speed with regard to the *observer* depends on whether the latter is moving with regard to the medium.
The constancy of light only strictly holds in a vacuum, and it is a logical consequence of the fact that light can travel without a carrier medium as it carries itself (to be more precise, according to Maxwell's Equations, the electric wave carries the magnetic wave and vice versa; it is somewhat ironic that Maxwell himself did obviously not realize this as he believed in the ether theory and a positive outcome of the Michelson-Morley experiment). On the other hand, experiments like the Sagnac effect, which apparently contradict the invariance of c and suggest the presence of an absolute reference frame, could well be explained by the presence of the earth's magnetic fieldSee also my page Speed of Light and Theory of Relativity for more.