We present new observations of very faint white dwarfs (WDs) in the rich open star cluster NGC 2099 (M37). Following deep, wide-field imaging of the cluster using the Canada-France-Hawaii Telescope, we have now obtained spectroscopic observations of candidate WDs using both the Gemini Multi-Object Spectrograph on Gemini North and the Low-Resolution Imaging Spectrometer on Keck. Of our 24 WD candidates (all fainter than V=22.4), 21 are spectroscopically confirmed to be bona fide WDs, four or five of which are most likely field objects. Fitting 18 of the 21 WD spectra with model atmospheres, we find that most WDs in this cluster are quite massive (0.7-0.9 Msolar), as expected given the cluster's young age (650 Myr) and, hence, high turnoff mass (~2.4 Msolar). We determine a new initial-final mass relationship and almost double the number of existing data points from previous studies. The results indicate that stars with initial masses between 2.8 and 3.4 Msolar lose 70%-75% of their mass through stellar evolution. For the first time, we find some evidence of a metallicity dependence on the initial-final mass relationship.Based on observations with Gemini (run ID GN-2002B-Q-11) and Keck. Gemini is an international partnership managed by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the National Science Foundation. The W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA, was made possible by the generous financial support of the W. M. Keck Foundation.