We study communication over the scalar Gaussian fading channel subject to a state-obfuscation constraint, which requires that the channel outputs and the fading coefficients be almost independent. We consider two cases for the fading coefficient: where it is independent and identically distributed in time, and where it is quasistatic, i.e., it is randomly generated but then remains the same during communication. For the transmitter, we study three different scenarios: where it only has access to the message; where it has channel-state information about the fading coefficient; and where it has access to feedback. We establish conditions for the communication capacity subject to obfuscation to be non-zero, and analyze this capacity in the high signal-to-noise ratio regime.