We have used the NASA Infrared Telescope Facility (IRTF) to observe the nuclear stellar cluster in the nearby, face-on, giant Scd spiral IC 342. From high-resolution (lambda/Delta lambda = 21,500) spectra of the (CO)-C-12 (2-0) band head at 2.3 mu m, we derive a line-of-sight stellar velocity dispersion sigma = (33 +/- 3) km s(-1). To interpret this observation we construct dynamical models based on the Jeans equation for a spherical system. The light distribution of the cluster is modeled using an archival Hubble Space Telescope (HST) V-band image and a new ground-based K-band image. Under the assumption of an isotropic velocity distribution, the observed kinematics imply a K-band mass-to-light ratio M/L-K = 0.05, and a cluster mass M approximate to 6 x 10(6) M-circle dot. We compare the derived mass-to-light ratio with the "Starburst99" stellar population synthesis models of Leitherer and collaborators and infer a best-fitting cluster age in the range 10(6.8-7.8) yr. Although this result depends somewhat on a number of uncertainties in the modeling (e.g., the assumed extinction along the line of sight toward the nucleus, the initial mass function of the stellar population model, and the velocity dispersion anisotropy of the cluster), none of the model parameters can be plausibly modified to yield a significantly larger age. Also, the inferred age is consistent with that found in our previous study based on the near-infrared absorption-line equivalent widths of the cluster (Boker, Forster-Schreiber, & Genzel). Recent HST observations of large samples of spiral galaxies have shown that nuclear stellar clusters are very common in intermediate- to late-type spirals. The cluster in IC 342 is more luminous than the clusters found in most other nearby spiral galaxies. If the nuclear stellar clusters in spiral galaxies all have a mass similar to that of the cluster in IC 342, then stellar population synthesis models indicate a median age for these clusters of several Gyr. This may be consistent with a scenario in which each spiral galaxy has only one episode of nuclear star cluster formation. On the other hand, the incidence of young nuclear star clusters may be high enough to indicate that the formation of these clusters is a recurring phenomenon. Age and population studies for a larger sample of galaxies are necessary to distinguish between these scenarios and to determine how these nuclear stellar clusters are related to the secular evolution of their environment. As a by-product of our analysis, we infer that IC 342 cannot have any central black hole more massive than 5 x 10(5) M-circle dot. This is similar to 6 times less massive than the black hole inferred to exist in our Galaxy, consistent with the accumulating evidence that galaxies with less massive bulges harbor less massive black holes.
