NGC 3621 is a late-type (Sd) spiral galaxy with an active nucleus, previously detected through mid-infrared [Ne V] line emission. Archival Hubble Space Telescope (HST) images reveal that the galaxy contains a bright and compact nuclear star cluster. We present a new high-resolution optical spectrum of this nuclear cluster, obtained with the ESI Spectrograph at the Keck Observatory. The nucleus has a Seyfert 2 emission-line spectrum at optical wavelengths, supporting the hypothesis that a black hole is present. The line-of-sight stellar velocity dispersion of the cluster is sigma=43+/-3 km/s, one of the largest dispersions measured for any nuclear cluster in a late-type spiral galaxy. Combining this measurement with structural parameters measured from archival HST images, we carry out dynamical modeling based on the Jeans equation for a spherical star cluster containing a central point mass. The maximum black hole mass consistent with the measured stellar velocity dispersion is 3*10^6 solar masses. If the black hole mass is small compared with the clusters stellar mass, then the dynamical models imply a total stellar mass of ~1*10^7 solar masses, which is consistent with rough estimates of the stellar mass based on photometric measurements from HST images. From structural decomposition of 2MASS images, we find no clear evidence for a bulge in NGC 3621; the galaxy contains at most a very faint and inconspicuous pseudobulge component (M_K>-17.6 mag). NGC 3621 provides one of the best demonstrations that very late-type spirals can host both active nuclei and nuclear star clusters, and that low-mass black holes can occur in disk galaxies even in the absence of a substantial bulge.
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