Context. Evolved low-mass stars lose a significant fraction of their mass through stellar winds. While the overall morphology of the stellar wind structure during the asymptotic giant branch (AGB) phase is thought to be roughly spherically symmetric, the morphology changes dramatically during the post-AGB and planetary nebula phase, during which bipolar and multi-polar structures are often observed. Aims. We aim to study the inner wind structure of the closest well-known AGB star CW Leo. Different diagnostics probing different geometrical scales have implied a non-homogeneous mass-loss process for this star: dust clumps are observed at milli-arcsec scale, a bipolar structure is seen at arcsecond-scale, and multi-concentric shells are detected beyond 1 . Methods. We present the first ALMA Cycle 0 band 9 data around 650 GHz (450 μm) tracing the inner wind of CW Leo. The fullresolution data have a spatial resolution of 0. 42 × 0. 24, allowing us to study the morpho-kinematical structure of CW Leo within ∼6 . Results. We have detected 25 molecular emission lines in four spectral windows. The emission of all but one line is spatially resolved. The dust and molecular lines are centered around the continuum peak position, which is assumed to be dominated by stellar emission. The dust emission has an asymmetric distribution with a central peak flux density of ∼2 Jy. The molecular emission lines trace different regions in the wind acceleration region and imply that the wind velocity increases rapidly from about 5 R , almost reaching the terminal velocity at ∼11 R . The images prove that vibrational lines are excited close to the stellar surface and that SiO is a parent molecule. The channel maps for the brighter lines show a complex structure; specifically, for the 13 CO J = 6-5 line, different arcs are detected within the first few arcseconds. The curved structure in the position-velocity (PV) map of the 13 CO J = 6-5 line can be explained by a spiral structure in the inner wind of CW Leo, probably induced by a binary companion. From modelling the ALMA data, we deduce that the potential orbital axis for the binary system lies at a position angle of ∼10-20• to the north-east and that the spiral structure is seen almost edge-on. We infer an orbital period of 55 yr and a binary separation of 25 au (or ∼8.2 R ). We tentatively estimate that the companion is an unevolved low-mass main-sequence star. Conclusions. A scenario of a binary-induced spiral shell can explain the correlated structure seen in the ALMA PV images of CW Leo. Moreover, this scenario can also explain many other observational signatures seen at different spatial scales and in different wavelength regions, such as the bipolar structure and the almost concentric shells. ALMA data hence for the first time provide the crucial kinematical link between the dust clumps seen at milli-arcsecond scale and the almost concentric arcs seen at arcsecond scale.