  In an attempt to use Cepheid variables to determine the distance to the
Centaurus cluster, we have obtained images of NGC 4603 with the Hubble Space
Telescope on 9 epochs using WFPC2 and the F555W and F814W filters. This galaxy
has been suggested to lie within the ``Cen30'' portion of the cluster and is
the most distant object for which this method has been attempted. Previous
distance estimates for Cen30 have varied significantly and some have presented
disagreements with the peculiar velocity predicted from redshift surveys,
motivating this investigation. Using our observations, we have found 61
candidate Cepheid variable stars; however, a significant fraction of these
candidates are likely to be nonvariable stars whose magnitude measurement
errors happen to fit a Cepheid light curve of significant amplitude for some
choice of period and phase. Through a maximum likelihood technique, we
determine that we have observed 43 +/- 7 real Cepheids and that NGC 4603 has a
distance modulus of 32.61 +0.11/-0.10 (random, 1 sigma) +0.24/-0.25
(systematic, adding in quadrature), corresponding to a distance of 33.3 Mpc.
This is consistent with a number of recent estimates of the distance to NGC
4603 or Cen30 and implies a small peculiar velocity consistent with predictions
from the IRAS 1.2 Jy redshift survey if the galaxy lies in the foreground of
the cluster.
