To study the interstellar medium and the interplay between the atomic and molecular components in a low-metallicity environment, we present a complete high angular and spectral resolution map and position-position-velocity data cube of the 12CO(J = 2-1) emission from the Local Group galaxy Messier 33. Its metallicity is roughly half-solar, such that we can compare its interstellar medium with that of the Milky Way with the main changes being the metallicity and the gas mass fraction. The data have a 12″ angular resolution (~50 pc) with a spectral resolution of 2.6 km s-1 and a mean and median noise level of 20 mK per channel in antenna temperature. A radial cut along the major axis was also observed in the 12CO(J = 1-0) line. The CO data cube and integrated intensity map are optimal when using H i data to define the baseline window and the velocities over which the CO emission is integrated. Great care was taken when building these maps, testing different windowing and baseline options, and investigating the effect of error beam pickup. The total CO(2-1) luminosity is 2.8 × 107 K km s-1 pc2, following the spiral arms in the inner disk, with an average decrease in intensity approximately following an exponential disk with a scale length of 2.1 kpc. There is no clear variation in the CO(2-1/1-0) intensity ratio with radius and the average value is roughly 0.8. The total molecular gas mass is estimated, using a N(H2) /ICO(1 – 0) = 4 × 1020cm-2/(K km s-1) conversion factor, to be 3.1 × 108 M⊙, including helium. The CO spectra in the cube were shifted to zero velocity by subtracting the velocity of the H i peak from the CO spectra. Stacking these spectra over the whole disk yields a CO line with a half-power width of 12.4 km s-1. As a result, the velocity dispersion between the atomic and molecular components is extremely low, independently justifying the use of the H i line in building our maps. Stacking the spectra in concentric rings shows that the CO linewidth and possibly the CO-H i velocity dispersion decrease in the outer disk. The error beam pickup could produce the weak CO emission apparently from regions in which the H i line peak does not reach 10 K, such that no CO is actually detected in these regions. Using the CO(2-1) emission to trace the molecular gas, the probability distribution function of the H2 column density shows an excess at high column density above a log-normal distribution.
