We measure apparent velocities (v_app) of the Halpha and Hbeta Balmer line cores for 449 non-binary thin disk normal DA white dwarfs (WDs) using optical spectra taken for the ESO SN Ia Progenitor surveY (SPY/ Napiwotzki et al. 2001). Assuming these WDs are nearby and co-moving, we correct our velocities to the Local Standard of Rest so that the remaining stellar motions are random. By averaging over the sample, we are left with the mean gravitational redshift, : we find = = 32.57 +/- 1.17 km/s. Using the mass-radius relation from evolutionary models, this translates to a mean mass of 0.647 +0.013 -0.014 Msun. We interpret this as the mean mass for all DAs. Our results are in agreement with previous gravitational redshift studies but are significantly higher than all previous spectroscopic determinations except the recent findings of Tremblay & Bergeron (2009). Since the gravitational redshift method is independent of surface gravity from atmosphere models, we investigate the mean mass of DAs with spectroscopic Teff both above and below 12000 K/ fits to line profiles give a rapid increase in the mean mass with decreasing Teff. Our results are consistent with no significant change in mean mass: ^hot = 0.640 +/- 0.014 Msun and ^cool = 0.686 +0.035 -0.039 Msun.