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 CFHT, we have now obtained spectroscopic observations of candidate WDs using both GMOS on Gemini and LRIS on Keck. Of our 24 WD candidates (all fainter than V = 22.4), 21 are spectroscopically confirmed to be bona fide WDs, 4-5 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 Msun), as expected given the cluster's young age (650 Myr) and, hence, high turnoff mass (~2.4 Msun). 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 Msun 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.