The rate-and-state capacity of a state-dependent channel with a state-cognizant encoder is the highest possible rate of communication over the channel when the decoder—in addition to reliably decoding the data—must also reconstruct the state sequence with some required fidelity. Feedback from the channel output to the encoder is shown to increase this capacity even for channels that are memoryless with memoryless states. This capacity is calculated here for such channels with feedback when the state reconstruction fidelity is measured using a single-letter distortion function and the state sequence is revealed to the encoder in one of two different ways: strictly–causally or causally. For the noncausal case, we provide bounds on the capacity and identify a condition under which the bounds coincide. Feedback does not increase the rate-and-state capacity when the decoder must reconstruct the state sequence perfectly or, in some settings, when the channel is Gaussian and fidelity is measured in terms of mean squared-error.