The Carnegie Hubble Program (CHP) is designed to calibrate the extragalactic distance scale using data from the post-cryogenic era of the Spitzer Space Telescope. The ultimate goal of the CHP is a systematic improvement in the distance scale leading to a determination of the Hubble constant to within an accuracy of 2%. This paper focuses on the measurement and calibration of the Galactic Cepheid period-luminosity (PL, Leavitt) relation using the warm Spitzer/IRAC 1 and 2 bands at 3.6 and 4.5{mu}m. We present photometric measurements covering the period range 4-70 days for 37 Galactic Cepheids. Data at 24 phase points were collected for each star. Three PL relations of the form M=a(log(P)-1)+b are derived. The method adopted here takes the slope a to be -3.31, as determined from the Spitzer Large Magellanic Cloud (LMC) data of Scowcroft et al. (Cat. J/ApJ/743/76). Using the geometric Hubble Space Telescope guide-star distances to 10 Galactic Cepheids, we find a calibrated 3.6{mu}m PL zero point of -5.80+/-0.03. Together with our value for the LMC zero point, we determine a reddening-corrected distance modulus of 18.48+/-0.04mag to the LMC. The mid-IR period-color diagram and the [3.6]-[4.5] color variation with phase are interpreted in terms of CO absorption at 4.5{mu}m. This situation compromises the use of the 4.5{mu}m data for distance determinations.