A primary new capability of JWST is the ability to penetrate the dust in star-forming galaxies to identify and study the properties of young star clusters that remain embedded in dust and gas. In this Letter we combine new infrared images taken with JWST with our optical Hubble Space Telescope (HST) images of the starbursting barred (Seyfert2) spiral galaxy NGC 1365. We find that this galaxy has the richest population of massive young clusters of any known galaxy within 30 Mpc, with similar to 30 star clusters that are more massive than 10(6) M (circle dot) and younger than 10 Myr. Sixteen of these clusters are newly discovered from our JWST observations. An examination of the optical images reveals that 4 of 30 (similar to 13%) are so deeply embedded that they cannot be seen in the Hubble I band (A ( V ) greater than or similar to 10 mag), and that 11 of 30 (similar to 37%) are missing in the HST B band, so age and mass estimates from optical measurements alone are challenging. These numbers suggest that massive clusters in NGC 1365 remain completely obscured in the visible for similar to 1.3 +/- 0.7 Myr and are either completely or partially obscured for similar to 3.7 +/- 1.1 Myr. We also use the JWST observations to gain new insights into the triggering of star cluster formation by the collision of gas and dust streamers with gas and dust in the bar. The JWST images reveal previously unknown structures (e.g., bridges and overshoot regions from stars that form in the bar) that help us better understand the orbital dynamics of barred galaxies and associated star-forming rings. Finally, we note that the excellent spatial resolution of the NIRCAM F200W filter provides a better way to separate barely resolved compact clusters from individual stars based on their sizes.