Measurements of flux density are described for five planets, Mars, Jupiter, Saturn, Uranus, and Neptune, across the six Planck High Frequency Instrument frequency bands (100-857 GHz) and these are then compared with models and existing data. In our analysis, we have also included estimates of the brightness of Jupiter and Saturn at the three frequencies of the Planck Low Frequency Instrument (30, 44, and 70 GHz). The results provide constraints on the intrinsic brightness and the brightness time-variability of these planets. The majority of the planet flux density estimates are limited by systematic errors, but still yield better than 1% measurements in many cases. Applying data from Planck HFI, the Wilkinson Microwave Anisotropy Probe (WMAP), and the Atacama Cosmology Telescope (ACT) to a model that incorporates contributions from Saturn's rings to the planet's total flux density suggests a best fit value for the spectral index of Saturn's ring system of βring = 2.30 ± 0.03 over the 30-1000 GHz frequency range. Estimates of the polarization amplitude of the planets have also been made in the four bands that have polarization-sensitive detectors (100-353 GHz); this analysis provides a 95% confidence level upper limit on Mars's polarization of 1.8, 1.7, 1.2, and 1.7% at 100, 143, 217, and 353 GHz, respectively. The average ratio between the Planck-HFI measurements and the adopted model predictions for all five planets (excluding Jupiter observations for 353 GHz) is 1.004, 1.002, 1.021, and 1.033 for 100, 143, 217, and 353 GHz, respectively. Model predictions for planet thermodynamic temperatures are therefore consistent with the absolute calibration of Planck-HFI detectors at about the three-percent level. We compare our measurements with published results from recent cosmic microwave background experiments. In particular, we observe that the flux densities measured by Planck HFI and WMAP agree to within 2%. These results allow experiments operating in the mm-wavelength range to cross-calibrate against Planck and improve models of radiative transport used in planetary science.