We present a high-angular resolution study of the cometary-shaped ultracompact H II region G29.96-0.02. We have obtained similar to 10 " angular-resolution millimeter-wave maps of the region in transitions of (CO)-C-13, (CO)-O-18, CH3CN, CH3OH, and CS with the BIMA interferometer. We combine these data with complementary single-dish data of the (CO)-C-13, (CO)-O-18, and CS lines taken with the FCRAO 14 meter telescope. These data are compared with near-infrared JHK-band images with less than or equal to 0."19 angular resolution obtained with the Calar-Alto 3.5 m telescope. The (CO)-C-13 data show emission extended over a 3 x 2 pc region; however, the emission is strongly peaked near the head of the H II region. Strong CS, (CO)-O-18, and CH3CN emissions peak near the same location. The CH3CN (J = 6 --> 5) emission peaks toward the hot core previously detected in VLA NH3(4, 4) observations, and we determine a kinetic temperature of 100 K in the core using a large velocity gradient analysis of the CH3CN (6 --> 5) BIMA data and CH3CN/(CH3CN)-C-13 (5 --> 4) IRAM 30 m telescope data. We also find that the sharply peaked (CO)-O-18, (CO)-C-13, and CS emission is indicative of a density gradient, with the peak density located in front of the head of the cometary H II region. We use our near-infrared data to search for sources embedded in the H II region and the adjacent cloud. In addition to the exciting star of the H II region, we identify a second star toward the head of the H II region with an extinction similar to that of the exciting star; this appears to be a second OB star in the H II region. Directly in front of the H II region we detect a highly reddened source, which is most likely a young star deeply embedded in the molecular gas. Furthermore, we find an enhanced density of sources with H-K >1 toward the molecular cloud and argue that these sources form an embedded cluster. Finally, we compare our results with current models of cometary shaped H II regions. Given the evidence that the G29.96 H II region exists in a gradient of molecular gas density that peaks in front of the head of the H II region, we favor the champagne flow model for this region. Comparing the measured densities, temperatures, and line widths of the ionized and molecular gas, we estimate the expansion speed of the H II region into the molecular core at 2-5 km s(-1).