We present the first grating-resolution X-ray spectra of the Seyfert 1 galaxy NGC 3783, obtained with the High Energy Transmission Grating Spectrometer on the Chandra X-Ray Observatory. These spectra reveal many narrow absorption lines from the H-like and He-like ions of O, Ne, Mg, Si, S, and Ar as well as Fe xvii-Fe xxi L-shell lines. We have also identified several weak emission lines, mainly from O and Ne. The absorption lines are blueshifted by a mean velocity of approximately 440+/-200 km s-1 and are not resolved, indicating a velocity dispersion within the absorbing gas of a few hundred kilometers per second or less. We measure the lines' equivalent widths and compare them with the predictions of photoionization models. The best-fitting model has a microturbulence velocity of 150 km s-1 and a hydrogen column density of 1.3x1022 cm-2. The measured blueshifts and inferred velocity dispersions of the X-ray absorption lines are consistent with those of the strongest UV absorption lines observed in this object. However, simple models that propose to strictly unify the X-ray and UV absorbers have difficulty explaining simultaneously the X-ray and UV absorption-line strengths.