  We present the results from a 42 ks \chandra ACIS observation of the galaxy
cluster A478. This cluster is generally considered to be highly relaxed. The
\chandra image reveals, for the first time, X-ray cavities in the hot gas
within the central 15 kpc radius of A478. Two weak and small ($\sim$ 4 kpc)
radio lobes that extend from the central nucleus, are detected in a 1.4 GHz VLA
observation. The radio lobes are roughly along the direction of the X-ray
cavities, but are much smaller than the X-ray cavities. We propose a ``donut''
configuration for the hot gas within the central 15 kpc, created by the
interaction of the gas with the radio plasma that originated from the nucleus.
The current radio activity of the central radio source is weak ($\sim$ 0.2% of
Hydra A) and the total radio power is at least 10 times smaller than the
minimum power needed to create the cavities. We compare A478 with other galaxy
clusters where similar X-ray cavities were found. A478 and A4059 host much
weaker central radio sources than do others with similar size X-ray cavities.
On larger scales, deprojected temperature and density profiles are obtained for
A478. We used these to derive the Compton y-parameter, for the first time,
through direct integration. The result has a much smaller statistical
uncertainty than previous ones. This serves as an example of how high quality
X-ray data can help constrain H$_{0}$. The corresponding H$_{0}$ also was
derived combining the available Sunyaev-Zeldovich effect (SZE) measurement.
