Although single units in primary auditory cortex (A1) exhibit accurate timing in their phasic response to the onset of sound (precision of a few milliseconds), paradoxically, they are unable to sustain synchronized responses to repeated stimuli at rates much beyond 20 Hz. To explore the relationship between these two aspects of cortical response, we designed a broadband stimulus with a slowly modulated spectrotemporal envelope riding on top of a rapidly modulated waveform (or fine structure). Using this stimulus, we quantified the ability of cortical cells to encode independently and simultaneously the stimulus envelope and fine structure. Specifically, by reverse-correlating unit responses with these two stimulus dimensions, we measured the spectrotemporal response fields (STRFs) associated with the processing of the envelope, the fine structure, and the complete stimulus. A1 cells respond well to the slow spectrotemporal envelopes and produce a wide variety of STRFs. In over 70% of cases, A1 units also track the fine-structure modulations precisely, throughout the stimulus, and for frequencies up to several hundred Hertz. Such a dual response, however, is contingent on the cell being driven by both fast and slow modulations, in that the response to the slowly modulated envelope gates the expression of the fine structure. We also demonstrate that either a simplified model of synaptic depression and facilitation, and/or a cortical network of thalamic excitation and cortical inhibition can account for major trends in the observed findings. Finally, we discuss the potential functional significance and perceptual relevance of these coexistent, complementary dynamic response modes.