We present the first detailed photometric and spectroscopic study of the white dwarfs (WDs) in the field of the ~225 Myr old (log tau_cl = 8.35) open cluster NGC 1039 (M34) as part of the ongoing Lick-Arizona White Dwarf Survey. Using wide-field UBV imaging, we photometrically select 44 WD candidates in this field. We spectroscopically identify 19 of these objects as WDs; 17 are hydrogen-atmosphere DA WDs, one is a helium-atmosphere DB WD, and one is a cool DC WD that exhibits no detectable absorption lines. We find an effective temperature (T_eff) and surface gravity (log g) for each DA WD by fitting Balmer-line profiles from model atmospheres to the observed spectra. WD evolutionary models are then invoked to derive masses and cooling times for each DA WD. Of the 17 DAs, five are at the approximate distance modulus of the cluster. Another WD with a distance modulus 0.45 mag brighter than that of the cluster could be a double-degenerate binary cluster member, but is more likely to be a field WD. We place the five single cluster member WDs in the empirical initial-final mass relation and find that three of them lie very close to the previously derived linear relation; two have WD masses significantly below the relation. These outliers may have experienced some sort of enhanced mass loss or binary evolution; however, it is quite possible that these WDs are simply interlopers from the field WD population. Eight of the 17 DA WDs show significant CaII K absorption; comparison of the absorption strength with the WD distances suggests that the absorption is interstellar, though this cannot be confirmed with the current data. ; Comment: 24 pages, 13 figures. Accepted for publication in the Astronomical Journal. Figures 1, 2 and 3 reduced in resolution