Our 50 ks Chandra observation of the small radio supernova remnant (SNR) G1.9+0.3 shows a complete shell structure with strong bilateral symmetry, about 100'' in diameter. The radio morphology is also shell-like, but only about 84'' in diameter, based on observations made in 1985. We attribute the size difference to expansion between 1985 and our Chandra observations of 2007. Expansion is confirmed in comparing radio images from 1985 and 2008. We deduce that G1.9+0.3 is of order 100 years old—the youngest supernova remnant in the Galaxy. Based on a very high absorbing column density of 5.5 × 10²² cm⁻², we place G1.9+0.3 near the Galactic center, at a distance of about 8.5 kpc, where the mean remnant radius would be about 2 pc, and the required expansion speed about 14,000 km s⁻¹. The X-ray spectrum is featureless and well described by the exponentially cut off synchrotron model srcut. With the radio flux at 1 GHz fixed at 0.9 Jy, we find a spectral index of 0.65 and a rolloff frequency of 1.4 × 10¹⁸ Hz. The implied characteristic rolloff electron energy of about 94(B/10 μ G)^−1/2 TeV is the highest ever reported for a shell supernova remnant. It can easily be reached by standard diffusive shock acceleration, given the very high shock velocities; it can be well described by either age-limited or synchrotron-loss-limited acceleration. Not only is G1.9+0.3 the youngest known Galactic remnant, it is also only the fourth Galactic X-ray-synchrotron-dominated shell supernova remnant.