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It has long been know that an extremely steep spectrum radio source lies atthe center of ZwCl 0735.7+7421, a galaxy cluster with high X-ray luminosity anda cooling core. In this Letter, we present VLA observations of this radiosource at both 1425 and 325 MHz. With a resolution below 21'' (75 kpc) for both1425 and 325 MHz, we show the morphology of the central source, and find thatit is most likely a large (400 kpc) radio galaxy rather than diffuse clusteremission. We estimate a steep spectral index of $\alpha_{325}^{1400} = -1.54$for the core (although it may be contaminated by lobe emission), while theouter lobes are extremely steep objects, both with $\alpha_{325}^{1400} <-3.1$. We also find evidence for restarted core activity in the form of a setof inner lobes oriented at a somewhat different angle from the outer lobes. Aspectral analysis extending the frequency range down to 74 MHz appears to showa turnover at very low frequencies. Comparison of the minimum energy radiopressures with the average thermal pressure surrounding the radio cavities fromMcNamara et al.(2005) shows that the radio lobes appear to be roughly inpressure balance with the thermal gas.Comment: Accepted for publication in the Astrophysical Journal Letters; 12  pages, 5 figures, 2 table

Resolving the Steep-Spectrum Emission in the Central Radio Source in ZwCl 0735.7+7421