In order to study the ISM 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 data cube of the 12CO(2-1) emission from the Local Group galaxy M33. Its metallicity is roughly half-solar, such that we can compare its ISM 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 (50pc) with a spectral resolution of 2.6 km/s and a mean noise level of 20 mK per channel in antenna temperature. A radial cut along the major axis was also observed in the 12CO(1-0) line. The CO data cube and integrated intensity map are optimal when using HI 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 2.8e7 K km/s pc2, following the spiral arms in the inner disk. 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)=4e20 cm-2/(K km/s) conversion factor, to be 3.1e8 Msol. The CO spectra in the cube were shifted to zero velocity by subtracting the velocity of the HI peak from the CO spectra. Hence, the velocity dispersion between the atomic and molecular components is extremely low, independently justifying the use of the HI line in building our maps. Stacking the spectra in concentric rings shows that the CO linewidth and possibly the CO-HI velocity dispersion decrease in the outer disk. 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.