Determining temperatures in molecular clouds from ratios of CO rotational lines or from ratios of continuum emission in different wavelength bands suffers from reduced temperature sensitivity in the high-temperature limit. In theory, the ratio of far-IR, submillimeter, or millimeter continuum to that of a ¹³CO (or C¹⁸O) rotational line can place reliable upper limits on the temperature of the dust and molecular gas. Consequently, far-infrared continuum data from the COBE/DIRBE instrument and Nagoya 4-m ¹³CO J = 1 → 0 spectral line data were used to plot 240 μm/¹³CO J = 1 → 0 intensity ratios against 140 μm/240 μm dust color temperatures, allowing us to constrain the multiparsec-scale physical conditions in the OrionA and B molecular clouds. The best-fitting models to the Orion clouds consist of two components: a component near the surface of the clouds that is heated primarily by a very large-scale (i.e. ∼ 1 kpc) interstellar radiation field and a component deeper within the clouds. The former has a fixed temperature and the latter has a range of temperatures that varies from one sightline to another. The models require a dust-gas temperature difference of 0±2K and suggest that 40-50% of the Orion clouds are in the form of dust and gas with temperatures between 3 and 10K. These results have a number implications that are discussed in detail in later papers. These include stronger dust-gas thermal coupling and higher Galactic-scale molecular gas temperatures than are usually accepted, an improved explanation for the N(H₂)/I(CO) conversion factor, and ruling out one dust grain alignment mechanism.