We have observed the H76-alpha recombination line toward the cometary ultracompact (UC) H II region G29.96-0.02 using the VLA. Our images have a resolution of 0.62" x 0.49" and an rms sensitivity of 1.87 mJy beam-1. They show a clear velocity gradient in the ionized gas which runs from approximately 80 km s-1 on the leading edge of the cometary structure to approximately 105 km s-1 in the "tail." The velocity gradient is steeper in front of cometary arc than behind it, and the line is approximately 40 km s-1 wide in the "head" while only approximately 25 km s-1 wide in the tail. Bulk motions of the ionized gas appear to have broadened the lines and produced steep velocity gradients. A comparison of our radio data with IR emission lines indicates that the visual extinction to G29.96-0.02 must be between 26 and 15 magnitudes, so the exciting star is clearly embedded in a molecular cloud. These observations are consistent with the bow-shock hypothesis for UC H II regions but inconsistent with current champagne flow models. We have evaluated the physical parameters of the ionized gas at five representative positions in the nebula and calculated non-LTE models of the line excitation. At present only the H76-alpha transition is available at high spatial resolution, making it impossible to obtain a unique solution. Non-LTE line models combined with the continuum image imply that electron densities in the arc are 5 to 10 times greater than in the tail of the nebula. The densities in the arc are typically about 8.5 x 10(4) cm-3, too low to cause significant electron impact broadening of the H76-alpha line. LTE electron temperatures are systematically low throughout the nebula, ranging from about 2500 K to 4200 K at different points in the H II region. Non-LTE models combined with additional constraints from the continuum image indicate that the electron temperature is about 5000 K throughout the nebula. This temperature is consistent with the galactocentric radius (4.2 kpc) of G29.96-0.02 and the galactic abundance gradient derived by Shaver et al. It is also consistent with the direct abundance measurements of Herter et al., Lacy, Beck, & Geballe; and Simpson & Rubin, who find that G29.96-0.02 is overabundant in O, Ne, Ar, and S.