We present the results of the first X-ray gratings spectroscopy observations of a planetary nebula (PN)—the X-ray-bright, young BD +30°3639. We observed BD +30°3639 for a total of ~300 ks with the Chandra X-ray Observatory's Low Energy Transmission Gratings in combination with its Advanced CCD Imaging Spectrometer (LETG/ACIS-S). The LETG/ACIS-S spectrum of BD +30°3639 is dominated by H-like resonance lines of O VIII and C VI and the He-like triplet line complexes of Ne IX and O VII. Other H-like resonance lines, such as N VII, and lines of highly-ionized Fe are weak or absent. Continuum emission is evident over the range 6-18 Å. Spectral modeling indicates the presence of a range of plasma temperatures from T_x ~ 1.7 × 10⁶ K to 2.9 × 10⁶ K and an intervening absorbing column N_H ~ 2.4 × 10²¹ cm^–2. The same modeling conclusively demonstrates that C and Ne are highly enhanced, with abundance ratios of C/O ~ 15-45 and Ne/O ~ 3.3-5.0 (90% confidence ranges, relative to the solar ratios), while N and Fe are depleted, with abundances N/O ~ 0.0-1.0 and Fe/O ~ 0.1-0.4, respectively. The intrinsic luminosity of the X-ray source determined from the modeling and the measured flux (F_X = 4.1 × 10^–13 ergs cm^–2 s^–1) is L_X ~ 8.6 × 10³² erg s^–1 (assuming D = 1.2 kpc). These gratings spectroscopy results are generally consistent with earlier results obtained from X-ray CCD imaging spectroscopy of BD +30°3639, but are far more precise. Hence, the Chandra/LETG-S results for BD +30°3639 place severe new constraints on models of PN wind-wind interactions in which X-ray emitting gas within PNs is generated via shocks and the plasma temperature is moderated by effects such as heat conduction or rapid evolution of the fast wind. The tight constraints placed on the (nonsolar) abundances directly implicate the present-day central star—hence, ultimately, the intershell region of the progenitor asymptotic giant branch star—as the origin of the shocked plasma now emitting in X-rays.