The absolute flux calibration of the James Webb Space Telescope will be based on a set of stars observed by the Hubble and Spitzer Space Telescopes. In order to cross-calibrate the two facilities, several A, G, and white dwarf (WD) stars are observed with both Spitzer and Hubble and are the prototypes for a set of JWST calibration standards. The flux calibration constants for the four Spitzer IRAC bands 1-4 are derived from these stars and are 2.3, 1.9, 2.0, and 0.5% lower than the official cold-mission IRAC calibration of Reach et al. (2005), i.e. in agreement within their estimated errors of ~2%. The causes of these differences lie primarily in the IRAC data reduction and secondarily in the SEDs of our standard stars. The independent IRAC 8 micron band-4 fluxes of Rieke et al. (2008) are about 1.5 +/- 2% higher than those of Reach et al. and are also in agreement with our 8 micron result.