We present spatially-resolved K- and L-band spectra (at spectral resolution R = 230 and R = 60, respectively) of MWC 419, a Herbig Ae/Be star. The data were obtained simultaneously with a new configuration of the 85-m baseline Keck Interferometer. Our observations are sensitive to the radial distribution of temperature in the inner region of the disk of MWC 419. We fit the visibility data with both simple geometric and more physical disk models. The geometric models (uniform disk and Gaussian) show that the apparent size increases linearly with wavelength in the 2-4 microns wavelength region, suggesting that the disk is extended with a temperature gradient. A model having a power-law temperature gradient with radius simultaneously fits our interferometric measurements and the spectral energy distribution data from the literature. The slope of the power-law is close to that expected from an optically thick disk. Our spectrally dispersed interferometric measurements include the Br gamma emission line. The measured disk size at and around Br gamma suggests that emitting hydrogen gas is located inside (or within the inner regions) of the dust disk.