We have investigated the large-scale morphology and properties of the molecular gas in the Gem OB1 cloud complex by mapping over 32 deg(2) (177 pc x 221 pc) of the complex in (CO)-C-12(J = 1-0) and (CO)-C-13(J = 1-0) at 50 '' sampling with QUARRY on the FCRAO 14 m telescope. The most striking characteristic of the molecular line images are the series of arc- and ring-shaped structures found on spatial scales from a few parsecs in diameter up to at least 35 pc. The morphology and in some instances the kinematics suggest that these features represent swept-up molecular material, most likely from expanding H II regions and wind blown bubbles. The kinetic temperatures and column densities of the molecular gas were derived from the (CO)-C-12 and (CO)-C-13 data using the LTE analysis. Most of the molecular gas was found to have kinetic temperatures of less than or similar to 10 K, and 50% of the mass of gas is contained in lines of sight with H-2 column densities less than or similar to 2 x 10(21) cm(-2). It was found that only 10% of the molecular mass is contained in lines of sight with column densities in excess of 10(22) cm(-2), and that these regions are found almost exclusively near the massive star forming regions within the arcs and rings of molecular gas. The average H-2 densities in areas with (CO)-C-13 emission are between 65-120 cm(-3), consistent with previous studies of cloud complexes, and is independent of whether the regions contains massive star formation or not. For the Gem OB1 complex as a whole, the average H-2 density is 1.2 cm(-3), which is only a few times the average atomic hydrogen density in the interstellar medium. We suggest an overall picture for the Gem OB1 complex in which most of the molecular gas is contained in relatively cold, low column density molecular material. The high column density regions in the Gem OB1 complex form through the external compression of the molecular gas by the winds and H II regions from newly formed massive stars. Thus once massive star formation is initiated, the structure and further evolution of the cloud complex is largely a result of the interactions of expanding H II regions and stellar winds with the ambient molecular material.