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New large-scale CO surveys of the first and second Galactic quadrants and the nearby molecular cloud complexes in Orion and Taurus, obtained with the CfA 1.2 m telescope, have been combined with 31 other surveys obtained over the past two decades with that instrument and a similar telescope on Cerro Tololo in Chile, to produce a new composite CO survey of the entire Milky Way. The survey consists of 488,000 spectra that Nyquist or beamwidth (1/8-deg) sample the entire Galactic plane over a strip 4-10 deg wide in latitude, and beamwidth or 1/4-deg sample nearly all large local clouds at higher latitudes. A map of molecular column density predicted from complete and unbiased far-infrared and 21 cm surveys of the Galaxy was used both to determine the completeness of the present survey and to extrapolate it to the entire sky at |b| < 32 deg. The close agreement of the observed and predicted maps implies that only ~2% of the total CO emission at |b| < 32 deg lies outside our current sampling. Taking into account this small amount of unobserved emission, the mean molecular column density decreases from ~3 E20 cm-2 at |b| = 5 deg to ~0.1 E20 cm-2 at |b| = 30 deg. The ratio of the predicted molecular column density map to the observed CO intensity map provides a calibration of the CO-to-H2 mass conversion factor X = N(H2)/Wco. Out of the Galactic plane (|b| > 5 deg), X shows little systematic variation with latitude from a mean value of 1.8 +/- 0.3 E20 cm-2 K-1 km-1 s. Given the large sky area and large quantity of CO data analyzed, we conclude that this is the most reliable measurement to date of the mean X value in the solar neighborhood.