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 for 9 epochs (totalling 24 orbits) over 14 months in the F555W filter and 2 epochs (totalling six orbits) in the F814W filter. This galaxy has been suggested to lie within the "Cen 30" portion of the Centaurus cluster, which is concentrated around a heliocentric redshift of approx3000 km s~(-1) and is the most distant object for which this method has been attempted. Previous distance estimates for Cen 30 have varied significantly, and some have presented disagreements with the peculiar velocity predicted on the basis of full-sky redshift surveys of galaxies, motivating our investigation. Using our WFPC2 observations, we have found 61 candidate Cepheid variable stars with well-determined oscillation periods and mean magnitudes, however, a significant frac- tion 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 (with zero excluded at greater than 9 #sigma#) and that NGC 4603 has a distance modulus of 32.61_(-0.10)~(+0.11) (random, 1 #sigma#)_(-0.25)~(+0.24)(systematic, adding in quadrature), corresponding to a distance of 33.3_(-1.5)~(+ 1.7) (random, 1 #sigma#)_(-3.7)~(+3.8) (systematic) Mpc. This result is consistent with a number of recent estimates of the distance to NGC 4603 or Cen 30
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