  It has long been know that an extremely steep spectrum radio source lies at
the center of ZwCl 0735.7+7421, a galaxy cluster with high X-ray luminosity and
a cooling core. In this Letter, we present VLA observations of this radio
source at both 1425 and 325 MHz. With a resolution below 21'' (75 kpc) for both
1425 and 325 MHz, we show the morphology of the central source, and find that
it is most likely a large (400 kpc) radio galaxy rather than diffuse cluster
emission. We estimate a steep spectral index of $\alpha_{325}^{1400} = -1.54$
for the core (although it may be contaminated by lobe emission), while the
outer 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 set
of inner lobes oriented at a somewhat different angle from the outer lobes. A
spectral analysis extending the frequency range down to 74 MHz appears to show
a turnover at very low frequencies. Comparison of the minimum energy radio
pressures with the average thermal pressure surrounding the radio cavities from
McNamara et al.(2005) shows that the radio lobes appear to be roughly in
pressure balance with the thermal gas.
