  _NGC 4589, a bright E2 merger-remnant galaxy, hosts the peculiar fast and
faint calcium-rich Type Ib supernova (SN) SN 2005cz. The progenitor of Ca-rich
SNe Ib has been controversial: it could be a) a young massive star with 6-12
M$\odot$ in a binary system, or b) an old low-mass star in a binary system that
was kicked out from the galaxy center. Moreover, previous distance estimates
for this galaxy have shown a large spread, ranging from 20 Mpc to 60 Mpc. Thus,
using archival $Hubble$ $Space$ $Telescope$/ACS $F435W$, $F555W$, and $F814W$
images, we search for star clusters in NGC 4589 in order to help resolve these
issues. We find a small population of young star clusters with $25<V\leq27$
($-7.1<M_V\leq-5.1$) mag and age $< 1$ Gyr in the central region at $R<0.5'$
($<3.8$ kpc), thus supporting the massive-star progenitor scenario for SN
2005cz. In addition to young star clusters, we also find a large population of
old globular clusters. In contrast to previous results in the literature, we
find that the color distribution of the globular clusters is clearly bimodal.
The turnover (Vega) magnitude in the $V$-band luminosity functions of the blue
(metal-poor) globular clusters is determined to be $V_0{(\rm
max)}=24.40\pm0.10$ mag. We derive the total number of globular clusters,
$N_{\rm GC} =640\pm50$, and the specific frequency, $S_N =1.7\pm0.2$. Adopting
a calibration for the metal-poor globular clusters, $M_V({\rm
max})=-7.66\pm0.14$ mag, we derive a distance to this galaxy:
$(m-M)_0=32.06\pm0.10({\rm ran})\pm0.15({\rm sys})$ ($d=25.8\pm2.2$ Mpc).
