During their late pulsating phase, AGB stars expel most of their mass in the form of massive dusty envelopes, an event that largely controls the composition of interstellar matter. The envelopes, however, are distant and opaque to visible and NIR radiation: their structure remains poorly known and the mass-loss process poorly understood. Millimeter-wave interferometry, which combines the advantages of longer wavelength, high angular resolution and very high spectral resolution is the optimal investigative tool for this purpose. Mm waves pass through dust with almost no attenuation. Their spectrum is rich in molecular lines and hosts the fundamental lines of the ubiquitous CO molecule, allowing a tomographic reconstruction of the envelope structure. The circumstellar envelope IRC +10 216 and its central star, the C-rich TP-AGB star closest to the Sun, are the best objects for such an investigation. Two years ago, we reported the first detailed study of the CO(2-1) line emission in that envelope, made with the IRAM 30-m telescope. It revealed a series of dense gas shells, expanding at a uniform radial velocity. The limited resolution of the telescope (HPBW 11″) did not allow us to resolve the shell structure. We now report much higher angular resolution observations of CO(2-1), CO(1-0), CN(2-1) and C4H(24-23) made with the SMA, PdB and ALMA interferometers (with synthesized half-power beamwidths of 3″, 1″ and 0.3″, respectively). Although the envelope appears much more intricate at high resolution than with an 11″ beam, its prevailing structure remains a pattern of thin, nearly concentric shells. The average separation between the brightest CO shells is 16″ in the outer envelope, where it appears remarkably constant. Closer to the star (< 40″), the shell pattern is denser and less regular, showing intermediary arcs. Outside the small (r < 0.3″) dust formation zone, the gas appears to expand radially at a constant velocity, 14.5 km s-1, with small turbulent motions. Based on that property, we have reconstructed the 3-D structure of the outer envelope and have derived the gas temperature and density radial profiles in the inner (r < 25″) envelope. The shell-intershell density contrast is found to be typically 3. The over-dense shells have spherical or slightly oblate shapes and typically extend over a few steradians, implying isotropic mass loss. The regular spacing of shells in the outer envelope supports the model of a binary star system with a period of 700 years and a near face-on elliptical orbit. The companion fly-by triggers enhanced episodes of mass loss near periastron. The densification of the shell pattern observed in the central part of the envelope suggests a more complex scenario for the last few thousand years.