We confirm the composite nature of the supernova remnant (SNR) G29.7-0.3 (Kes 75) using observations in the 0.5-8 keV energy range obtained with the ASCA X-ray observatory. Spatial and spectral analyses of the data show that G29.7-0.3 is composed of a hard X-ray, nonthermal central source and a soft X-ray, thermal shell. Combining the results of our analysis with radio determinations of the distance to G29.7-0.3 and its shell diameter (Becker & Helfand), we estimate the luminosities of the two components of the SNR as well as its total mass and age. The Crab-like core has an X-ray luminosity of L(x) approximate to 1 x 10(36) ergs s(-1) making it the second most luminous synchrotron nebula in the Galaxy. The thermal X-ray emission of the shell supports an age estimate comparable to that of the Crab, and the shell's inferred composition and mass are consistent with a massive progenitor. An estimate of the current energy loss rate of the unseen pulsar from the synchrotron nebula's X-ray luminosity, coupled with a calculation of the total energy in nebular particles and fields from the shape of the radio-to-Xray spectrum, also gives an age estimate of approximate to 10(3) yr.